Comparing Cartesian closed categories of (core) compactly generated spaces

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Topology and its Applications 143 (2004) 105–145

www.elsevier.com/locate/topo

Comparing Cartesian closed categoriesof (core) compactly generated spaces

Martín Escardóa, Jimmie Lawsonb,∗, Alex Simpsonc

a School of Computer Science, University of Birmingham, UKb Department of Mathematics, Louisiana State University, USA

c LFCS, School of Informatics, University of Edinburgh, UK

Received 17 October 2003; accepted 12 February 2004

Abstract

It is well known that, although the category of topological spaces is not Cartesian clospossesses many Cartesian closed full subcategories, e.g.: (i) compactly generated Hausdor(ii) quotients of locally compactHausdorff spaces, which form a larger category; (iii) quotients olocally compact spaces without separation axiom, which form an even larger one; (iv) quotiecore compact spaces, which is at least as large as the previous; (v) sequential spaces, wstrictly included in (ii); and (vi) quotients of countably based spaces, which are strictly includthe category (v).

We give a simple and uniform proof of Cartesian closedness for many categories of topospaces, including (ii)–(v), and implicitly (i), and we also give a self-contained proof that (vCartesian closed. Our main aim, however, is to compare the categories (i)–(vi), and others lik

When restricted to Hausdorff spaces, (ii)–(iv) collapse to (i), and most non-Hausdorff sof interest, such as those which occur in domain theory, are already in (ii). RegardinCartesian closed structure, finite products coincide in (i)–(vi). Function spaces are charactecoreflections of both the Isbell and natural topologies. In general, the function spaces differ bthe categories, but those of (vi) coincide with those in any of the larger categories (ii)–(v). Fthe topologies of the spaces in the categories (i)–(iv) are analyzed in terms of Lawson duality. 2004 Elsevier B.V. All rights reserved.

MSC:54D50; 54D55; 54C35; 06B35

Keywords:k-space; Compactly generated space; Function space; Isbell topology; Cartesian closed category;Quotient space; Core-compact space; Countably based space; Domain theory

* Corresponding author.E-mail address:[email protected] (J. Lawson).

0166-8641/$ – see front matter 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.topol.2004.02.011

106 M. Escardó et al. / Topology and its Applications 143 (2004) 105–145

1. Introduction

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Let X and Y be topological spaces and let C(X,Y ) denote the set of continuoumaps fromX to Y . Given any continuous mapf :A × X → Y , one has a functionf :A → C(X,Y ) defined byf (a) = (x → f (a, x)), called the exponentialtransposeof f .A topology on C(X,Y ) is said to beexponentialif continuity of a functionf :A× X → Y

is equivalent to that of its transposef :A → C(X,Y ). When an exponential topologexists, it is unique, and we denote the resulting space by[X ⇒ Y ]. This is elaboratedin Section 2 below. In this case, transposition is a bijection from the set C(A × X,Y ) tothe set C(A, [X ⇒ Y ]); this bijective equivalence, when it holds, is often referred to asexponential law.

A spaceX is said to beexponentiableif for every spaceY there is an exponentiatopology on C(X,Y ). Among Hausdorff spaces, the exponentiable ones are those that alocally compact. In general, a space is exponentiable if and only if it iscore compact;that is, every neighbourhoodV of a point contains a neighbourhoodU of the point suchthat every open cover ofV has a finite subcover ofU . Thus, not all topological spaceare exponentiable. In categorical terminology, the category Top of continuous matopological spaces fails to be Cartesian closed. Furthermore, ifX andY are exponentiablspaces, it is hardly ever the case that the exponential[X ⇒ Y ] is again exponentiable. Ifact, the full subcategory of exponentiable spaces also fails to be Cartesian closed.

Nonetheless, Top is known to possess several Cartesian closed full subcateincluding:

(i) Compactly generated Hausdorff spaces, also known ask-spaces or Kelley spaces—see, e.g., [33,22,4]. These are the Hausdorff spaces such that any subspacintersection with every compact subspace is closed is itself closed.

(ii) Compactly generated spaces without separation axiom [7,8,34]. These are charactized as the quotients of locally compact Hausdorff spaces and include the ones ofprevious category. We show that they also include most of the non-Hausdorff sthat occur in domain theory [15].

(iii) The quotients of locally compact spaces [8,34]. (The appropriate notion of locacompactness in the absence of the Hausdorff separation axiom is that every poa base of compact, not necessarily open, neighbourhoods.) This subcategory inthe previous, and Isbell showed that the inclusion is strict, by exhibiting aT1 exampleof a space which belongs to this but not to the previous [20].

(iv) The quotients of core compact spaces [8,34]. These of course include the spaces ofprevious subcategory, but it is not known whether the inclusion is strict. Howeveshow that the categories (ii)–(iv) collapse to (i) when restricted to Hausdorff space

(v) The sequential spaces; that is, those whose topologies are determined by sequenconvergence. These are strictly included in (ii).

(vi) The quotients of countably based spaces [27,24]. The spaces here are striincluded in those of the previous category. This category is particularly relevacomputability considerations, and, as the third author has reported recently [supports a great deal of programming-language semantics constructions.

M. Escardó et al. / Topology and its Applications 143 (2004) 105–145 107

(vii) The densely injective topological spaces [15]. These are strictly included in ii. (The

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injective topological spaces form yet another example, which is properly included inthis one, see [15].)

One of the contributions of this paper is to provide a simple and uniform pof Cartesian closedness for many subcategories of Top including (ii)–(v) (andimplicitly (i)). In addition, we give a proof that the category (vi) is Cartesian closHowever, the main goal of the paper is to compare the different categories.

It is well known that, for the category (vii), products and exponentials are calcuas in the category of topological spaces. It follows that products and exponentials apreserved by the inclusions of category (vii) in the categories (ii)–(vi). It is also knthat products in the categories (i)–(vi) again coincide with topological products inspecial cases, for example, when one of the factors is the unit interval (which is relevant fohomotopy theory). In general, however, products in the categories (i)–(vi) have topofiner than the topological product. Nevertheless, although, in principle, products mighwhen we move between categories (i)–(vi) (and others like them), we shall prove thfact, finite products are always calculated as in the largest category (iv).

Function spaces, in contrast, do vary betweenthe categories. However, we identify twimportant situations in which they remain the same: (i) For Hausdorff spaces, funspaces are calculated in the same way in each ofthe categories (i)–(iv). (ii) The functiospaces of (vi) are calculated in the same way as in any of its supercategories (ii)–order to prove these results, we derive useful general characterizations of function spacin the different categories.

Organization

In Section 2 we recall basic facts concerning exponentiability in the category otopological spaces. In Section 3 we give a simple proof of the Cartesian closednesscertain subcategories of Top, including the categories (ii)–(v) and implicitly (i). Sectshows that the Hausdorff objects of (ii)–(iv) coincide and that most non-Hausdorff spathat occur in domain theory belong to ii and hence to (iii) and (iv). Section 5 has a clook at products and function spaces in the categories (ii)–(v) and implicitly (i). Section 6contains various characterizations of the objects of (vi) in terms of pseudobasesis applied to investigate products and function spaces, in Section 7. Section 8 develcharacterizations of (some of) the objects of the categories in terms of Lawson dualiclose the paper with a few open problems, in Section 9.

2. Exponentiable topological spaces

In this preliminary section we recall known facts about function spaces of topologispaces which are needed for our purposes.

For setsA, X, Y , the natural bijectionf ↔ f between the sets of functions Set(A ×X,Y ) and Set(A,Set(X,Y )) associates withf :A × X → Y its transposef :A →Set(X,Y ) given by(f (a))(x) = f (a, x).


A topological spaceX is exponentiablein the category Top of topological spaces if for

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every spaceY there is a topology on the set C(X,Y ) of continuous maps fromX to Y

such that for any spaceA the associationf ↔ f is a bijection from the set of continuoumaps C(A× X,Y ) to the set of continuous maps C(A, [X ⇒ Y ]), where[X ⇒ Y ] denotesC(X,Y ) equipped with the hypothesized topology, which is referred to as anexponentialtopology. We often refer to this bijection as theexponential law. To maintain a categoricaperspective, we provisionally ignore the fact that these are known to be precisely thcompact spaces.

Lemma 2.1. For X exponentiable, the evaluation mapping

E : [X ⇒ Y ] × X → Y

defined byE(f,x) = f (x) is continuous.

Proof. The transpose ofE is the identity map on[X ⇒ Y ], hence continuous, and thusE

is continuous sinceX is exponentiable. Lemma 2.2. For X exponentiable, the exponential topology onC(X,Y ) is uniquelydetermined.

Proof. Suppose that[X ⇒ Y ] and[X ⇒′ Y ] are C(X,Y ) endowed with topologies thasatisfy the properties of an exponential. By Lemma 2.1 the evaluation mapE : [X ⇒Y ] × X → Y is continuous, and then the identity function on C(X,Y ) from [X ⇒ Y ] to[X ⇒′ Y ] is continuous since the latter is an exponential. Reversing the roles of[X ⇒ Y ]and[X ⇒′ Y ] gives continuity of the identity in the reverse direction.

In light of the preceding observation we denote by[X ⇒ Y ] the set C(X,Y ) ofcontinuous maps endowed with the exponential topology.

Lemma 2.3. For X exponentiable andg ∈ C(Y,Z), the map

[X ⇒ g] : [X ⇒ Y ] → [X ⇒ Z]defined by[X ⇒ g](f ) = g f is continuous.

Proof. The mapg E : [X ⇒ Y ] × X → Z is continuous by Lemma 2.1, and thustranspose, which one sees directly to be[X ⇒ g], is continuous.

It follows from the preceding that every exponentiableX gives rise to a functo[X ⇒ ·] : Top→ Top defined byY → [X ⇒ Y ] andg → [X ⇒ g]. One observes directlfrom the fact that[X ⇒ Y ] is an exponential that this functor is right adjoint to the func· × X.

Lemma 2.4. The product of two exponentiable spaces is exponentiable.


Proof. Let X1 andX2 be exponentiable spaces. Then for all spacesA, one has bijections

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C(A × (X1 × X2), Y ) ∼= C((A × X1) × X2, Y ) ∼= C(A × X1, [X2 ⇒ Y ]) ∼= C(A, [X1 ⇒[X2 ⇒ Y ]]). Further, C(X1, [X2 ⇒ Y ]) ∼= C(X1 ×X2, Y ). Hence the exponential topologon C(X1, [X2 ⇒ Y ]) induces an exponential topology on C(X1 × X2, Y ).

Given a familyXi of spaces, we endow the disjoint sum∑

i Xi with the final topology ofthe inclusionsXi → ∑

i Xi , that is, the finest topology making the inclusions continuoAs it is well known, this construction gives the categorical coproduct in Top.

Lemma 2.5. The disjoint sum of a family of exponentiable spaces is exponentiable.

Proof. Similar to the above. Show that ifXi is a family of exponentiable spaces andY isany space then

∏i[Xi ⇒ Y ] induces an exponentiable topology on C(

∑i Xi, Y ) via the

natural bijection∏

i C(Xi, Y ) ∼= C(∑

i Xi, Y ). A useful characterization of exponentiable spaces is the following.

Lemma 2.6. A spaceX is exponentiable if and only ifq × idX :A × X → B × X is aquotient map for every quotient mapq :A → B.

Proof. We provide the proof only in the direction that will be useful for our purpoat a later point—see Remark 2.7 below for a sketch of the other. Suppose thatX is anexponentiable space,q :A → B is a quotient map, andh :B × X → Y is a function suchthat

A × Xq×idX−→ B × X

h−→Y

is continuous. SinceX is exponentiable,

Aq−→B

h−→[X ⇒ Y ]is continuous, and thush is continuous, sinceq is quotient. Again using the exponentiabity of X, we conclude thath is continuous. It follows thatq × idX is quotient. Remark 2.7. A more categorical, and brief, proof of the direction provided wouldobserve that quotient maps are coequalizers and that left-adjoint functors preserve cThe other direction is also easy categorically speaking. It uses the adjoint-functor theThe solution-set condition is easy. One needs to check that the product functor· × X

preserves colimits. For this, it is enough to check the preservation of disjoint sumof coequalizers. The first is easy and the last is our assumption. See, e.g., Isbell [19].

The definition of exponentiability of a space quantifies over all topological spaces.finish this section with an intrinsic characterization.

Definition 2.8 (Core compact space). A topological spaceX is calledcore compactifevery open neighbourhoodV of a pointx of X contains an open neighbourhoodU of x

with the property that every open cover ofV has a finite subcover ofU .


For Hausdorff spaces, core compactnesscoincides with local compactness [15].

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Theorem 2.9. A space is exponentiable if and only if it is core compact.

Proof. In this generality, the theorem is essentially due to Day and Kelly [9]. We refereader to, e.g., Isbell [19], Chapter II.4 of [15] or the expository paper [13].

3. C-generated spaces

The purpose of this section is to provide a simple proof of a known theorem, sconditions under which the category ofC-generatedtopological spaces is Cartesian closWe begin by recalling basic notions and facts.

Definition 3.1 (C-generated space). Let C be a fixed collection of spaces, referred togenerating spaces. By aprobeover a spaceX we mean a continuous map from one of tgenerating spaces toX. TheC-generatedtopologyCX on a spaceX is the final topologyof the probes overX, that is, the finest topology making all probes continuous. Wethat a topological spaceX is C-generatedif X = CX. The category of continuous mapsC-generated spaces is denoted by TopC .

Notice that any collection generates the same class of spaces as the collection extewith the one-point space. Hence we shall assume without loss of generality thatC containsat least one non-empty space.

It is easily seen that, whenX is C-generated, a functionf :X → Y is continuous if andonly if f :X → CY is continuous. This means that TopC is a coreflective subcategoryTop (the so-calledcoreflective hullof C). The coreflection functor maps any continuof :X → Y to f :CX → CY , and the identity functionCY → Y gives the counit of thecoreflection. The lemma below summarizes basic properties of such coreflective hu

Lemma 3.2.

(i) Every generating space isC-generated.(ii) C-generated spaces are closed under the formation of quotients.(iii) C-generated spaces are closed under the formation of disjoint sums.(iv) AnyC-generated space is a quotient of a disjoint sum of generating spaces.(v) A space isC-generated if and only if it is a colimit inTop of generating spaces.

Proof. (i) Consider the identity probe.(ii) If f :X → Y is a quotient map andX is C-generated, then the composite prob

f p :C → Y suffice to generate the topology ofY , wherep :C → X varies over allprobes ofX.

(iii) Similar to that of (ii).(iv) Let X be aC-generated space andI be the set of non-open subsets ofX. By

definition of final topology, for eachi ∈ I , there exists a probepi :Ci → X with p−1i (i)


non-open. By the choice of probes, a subsetV of X is open if and only ifp−1(V ) isyepsss)

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open for alli ∈ I , which shows that the topology ofX coincides with the final topologof the family pi :Ci → Xi∈I . To ensure that all points ofX are covered by probes, wenlarge the familypi :Ci → Xi∈I slightly, if necessary, by including all constant mafrom some non-empty generating space. SinceI is a set (rather than just a proper clawe can take the disjoint sum of the spacesCi . Call it S and, for eachi ∈ I , let ji :Ci → S

be the inclusion. By the universal property of sums, there is a unique mapq :S → X suchthat q ji = pi for all i ∈ I . Again by the universal property, a functionf :X → Y iscontinuous if and only iff q :S → X is continuous, which shows thatq is a quotientmap.

(v) The construction of arbitrary small limits in any small cocomplete category (asis) can be reduced to coproducts (which in Top are disjoint sums) followed by coequa(which in Top are quotient maps).

We are particularly interested in the following four examples:

Definition 3.3 (Main examples). If C consists of respectively (i) core compact spaces,locally compact spaces, (iii) compact Hausdorff spaces, (iv) the one-point compactifiof the countable discrete space, then we refer toC-generated spaces as (i)core compactlygenerated spaces, (ii) locally compactly generated spaces, (iii) compactly generatespaces, (iv) sequentially generated spaces.

More examples are given at the end of this section.

Corollary 3.4.

(i) A space is core compactly generated if and only if it is a quotient of a core comspace.

(ii) A space is locally compactly generated if and only if it is a quotient of a loccompact space.

(iii) A space is compactly generated if and only if it is a quotient of a locally comHausdorff space.

(iv) A space is sequentially generated if and only if it is sequential.

Proof. This follows directly from Lemma 3.2 in light of the following observations.and (ii) The classes of core compact spacesand locally compact spaces are each clounder disjoint sums. (iii) A disjoint sum ofcompact Hausdorff spaces is locally compHausdorff. Conversely a locally compact Hausdorff space is compactly generated. (ifinal topology for probes on the one-point compactification of the countable discreteis the topology of sequential convergence.Definition 3.5 (Productive class of generating spaces). We say thatC is productiveif everygenerating space is exponentiable, and also the topological product of any two genspaces isC-generated.


Examples (i)–(iv) formulated in Definition 3.3 are all productive. In each case, theleirementcalroduct, is

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exponentiability requirement follows from the explicit characterization of exponentiabspaces as core compact spaces (Theorem 2.9). In cases (i)–(iii), the product requholds because the generating spaces are themselves closed under finite topologiproducts. In the case of example (iv), the condition is met because the topological pof any two countably based spaces, in particular of two copies of the generating spaceagain countably based, and hence sequential.

We now come to the main theorem concerningC-generated spaces, first provedDay [8].

Theorem 3.6. If C is productive thenTopC is Cartesian closed.

We give a proof which is simpler and more direct than that of [8].

Definition 3.7 (C-continuous map). We say that a mapf :X → Y of topological spaces iC-continuousif the compositef p :C → Y is continuous for every probep :C → X.

Of course, continuous maps areC-continuous. Moreover, for maps defined onC-generated spaces,C-continuity coincides with continuity.

Lemma 3.8 (The category MapC ).

(i) TheC-continuous maps of arbitrary topological spaces form a category, denoteMapC .

(ii) The identityCX → X is an isomorphism inMapC .(iii) The assignment that sends a spaceX to CX and aC-continuous map to itself is a

equivalence of categoriesC : MapC → TopC .

Proof. (i) The identity function is continuous, henceC-continuous. Letf :X → Y andg :Y → Z beC-continuous maps. In order to show thatg f :X → Z is C-continuous, letp :C → X be a probe. By theC-continuity off , the compositef p :C → Y is a probe,and by that ofg, so isg (f p) = (g f ) p, which shows thatg f is C-continuous.

(ii) Being continuous, the identityCX → X is C-continuous. By definition of the finatopology,p :C → CX is continuous for each probep :C → X, which, by definition, meanthat the identityX → CX is C-continuous.

(iii) TopC is a full subcategory of MapC , because a spaceX is C-generated if and onlyif continuity of a function defined onX is equivalent toC-continuity, and, as we have juseen, every space is isomorphic in MapC to an object of TopC . Lemma 3.9. Finite products in the categoryMapC exist and can be calculated as inTop.

Proof. We show that a topological productX1 × X2 has the required universal properThe projectionsπi :X1 × X2 → Xi , being continuous, areC-continuous. For each indexi,let fi :A → Xi beC-continuous, and letf :A → X1 × X2 be the unique (set-theoreticafunction withfi = πif . In order to show that it isC-continuous, letp :C → A be a probe


We have to show thatf p is continuous. By definition of topological product, this is

ducts,up

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a

equivalent to showing thatπi f p is continuous for each indexi. By construction, thisis the same asfip, which is continuous byC-continuity offi . Remark 3.10. Essentially the same proof shows thatall limits in MapC exist and can becalculated as in Top. We emphasize, however, that the limits, including the finite proare not uniquely determined as topologicalspaces up to homeomorphism, but onlyto isomorphism in MapC , where an isomorphism is aC-continuous bijection withC-continuous inverse.

Lemma 3.11. If f :X × Y → Z is a C-continuous map, then for eachx ∈ X, the functionfx :Y → Z defined byfx(y) = f (x, y) is C-continuous.

Proof. The functiony → (x, y) :Y → X × Y is continuous and henceC-continuous.Composition withf yieldsfx , which is thenC-continuous by Lemma 3.8(i).

For aC-continuous mapf :X × Y → Z, we thus have a functionf :X → MapC(Y,Z)

defined byf (x) = fx , where MapC(Y,Z) is the set ofC-continuous maps fromY to Z. Weshow that, whenC is productive and for MapC(Y,Z) suitably topologized, the functionfis C-continuous if and only if its transposef is, and thus establish the Cartesian closednof the category ofC-continuous maps.

In order to define the topology on MapC(Y,Z), we require all generating spaces toexponentiable. The topology is then constructed from the topologies of the expon[C ⇒ Z] in the category Top, whereC ranges overC. Each probep :C → Y induces afunctionTp : MapC(Y,Z) → [C ⇒ Z] defined byTp(g) = g p. We endow MapC(Y,Z)

with the initial topology of the family of functions that arise in this way, obtainingspace MapC[Y,Z]. By definition, this means that, for any spaceB, a functionh :B →MapC[Y,Z] is continuous if and only if the compositeTp h :B → [C ⇒ Z] is continuousfor each probep :C → Y .

Lemma 3.12. SupposeC is productive.

(i) A functionh :B → MapC[Y,Z] is continuous if and only if for each probep :C → Y ,the function(b, c) → (h(b))(p(c)) :B × C → Z is continuous.

(ii) The transposef :X → MapC[Y,Z] of a functionf :X × Y → Z is C-continuous ifand only if for all probesp :B → X andq :C → Y , the mapf (p × q) :B ×C → Z

is continuous.(iii) A functionf :X ×Y → Z is C-continuous if and only if for all probesp :B → X and

q :C → Y , the functionf (p × q) :B × C → Z is continuous.(iv) MapC is Cartesian closed.

Proof. (i) The compositeTp h :B → [C ⇒ Z] is the transpose of the function(b, c) →(h(b))(p(c)) :B × C → Z, and, by exponentiability of C in Top, a mapB → [C ⇒ Z] iscontinuous if and only if it is the transpose of a continuous mapB × C → Z.


(ii) By definition, f :X → MapC[Y,Z] is C-continuous if and only if the mapf e

bes

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p :B → MapC[Y,Z] is continuous for all probesp :B → X, and, by (i), this is the casif and only if the function(b, c) → ((f p)(b))(q(c)) :B × C → Z is continuous forall probesq :C → Y . The result then follows from the fact that((f p)(b))(q(c)) =f (p(b), q(c)) = f (p × q)(b, c).

(iii) ⇒: By productivity,B × C is C-generated, sof (p × q) :B × C → X × Z iscontinuous if and only iff (p × q) s is continuous for each probes :E → B × C. Butf (p × q) is C-continuous by Lemma 3.8(i), and hencef (p × q) s is continuous.

(iii) ⇐: Let r :D → X × Y be a probe. By composition with the projections, prop :D → X and q :D → Y are obtained. By hypothesis,f (p × q) :D × D → Z iscontinuous. And since the diagonal map∆ :D → D × D is continuous, so is the maf (p × q) ∆ = f r :D → Z.

(iv) By (ii) and (iii), C-continuity of a mapf :X × Y → Z is equivalent to that of itstransposef :X → MapC[Y,Z].

This and Lemma 3.8(iii) conclude our proof of Theorem 3.6. As already statedresult was first proved by Day [8]. Instead of working with our auxiliary category oC-continuous maps of arbitrary topological spaces, Day considered an enlarged categthereof that could readily be shown to be Cartesian closed, and he reflected this categonto the category ofC-generated spaces. Using fairly elaborate categorical machinery, Dainferred the Cartesian closedness of the category ofC-generated spaces, via an applicatof a general categorical reflection theorem. Our proof is considerably simpler becausedirectly obtain an equivalence rather than just a reflection.

Proofs more closely related to ours involving what we have called probes can bein Brown [5], [7, Chapter 5] and Vogt [34], although they restrict their attention to funcspaces that are given by the compact-open topology and itsC-coreflection. Barr [1] givesa general categorical argument for establishing themonoidal closureof categories, whichprovides another proof of Day’s theorem when specialized to categories of topolspaces.

Steenrod [33] is responsible for popularizing the category of compactly genespaces (alternativelyk-spaces) in the Hausdorff case, crediting Spanier [32], Wgram [35] and Brown [6], and referencing Kelley’s book [21]. The first reference into k-spaces seems to be Gale’s paper [14], whichattributes the notion and terminologyHurewicz, who gave seminars on thesubject at Princeton as early as 1948–1949.

Examples. It follows from Theorem 3.6 (together with the characterization of expontiable spaces as core compact spaces) that thefour examples introduced in Definition 3are Cartesian closed. Other examples are the following:

(1) Because discrete spaces are 1-generated, where 1 is the one-point space, they formCartesian closed category. This of courseamounts to the familiar fact that the categoof sets is Cartesian closed.

(2) An Alexandroff spaceis a space in which arbitrary intersections of open sets are oIt is an easy exercise to show that a space is Alexandroff if and only if it isS-generatedwhereS is the Sierpinski space, that is, the two-point space0,1 with open setsS,


1, and∅. Hence Alexandroff spaces form a Cartesian closed category. But it is wellps

[15].obernomain-ree formf

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known that Alexandroff spaces form a category isomorphic to that of monotone maof preordered sets, which is a familiar example of a Cartesian closed category.

(3) A domain is a continuous directed complete poset under the Scott topologyEquivalently, in purely topological terms, a domain is a locally supercompact sspace, where a space is calledsupercompactif every open cover has a singletosubcover. Because domains are closed under disjoint sums, a space is dgenerated if and only if it is a topological quotient of a domain. Because domains aexponentiable in Top and are closed under finite topological products, and henca productive class of exponentiable spaces, we conclude that topological quotients odomains form a Cartesian closed category.

4. (Core) compactly generated spaces

In this section we restrict our attention to the categories of core compactly generatcompactly generated spaces and consider some of their relationships (cf. DefinitioObviously, the smaller the setC of generating spaces, the finer the generated topoIn particular the core compactly generated topology is always contained in the comgenerated topology.

Our first observation will be that for Hausdorff spaces the two notions agree. In orderprove this, we introduce the relation ofrelative compactnessbetween arbitrary subsetsa topological space.

Definition 4.1 (The relation on sets). For arbitrary subsetsA andB of a topologicalspaceX, we say thatA is relatively compactin B, writtenA B, if for every open coveof B, there exist finitely many elements in the cover that coverA.

The notion of relative compactness, restricted to open sets, plays an important rthe “way below” relation) in the study of core compact spaces. Indeed, the definiticore compactness says thatX is core compact if, for every open neighbourhoodV of x,there exists an open neighbourhoodU of x with U V . Moreover, as is well known, thfollowing interpolation property holds [15].

Lemma 4.2 (Interpolation).If X is core compact andU,V ⊆ X are open withU V

then there exists openW ⊆ X such thatU W V .

The extension of the notion of relative compactness to arbitrary subsets generalizesfamiliar concept of compact subset. Indeed, a subsetA is compact if and only ifA A.Furthermore, basic topological results about compactness typically have analogurelative compactness. We give some examples, of which the first three are neededpurposes of this section and the last two are used in Section 7 below.

Lemma 4.3. LetX andY be topological spaces.

(i) If A is closed andA X, thenA is compact.


(ii) If A B ⊆ X andX is Hausdorff, thenA ⊆ B.

act,ct thatse, andhction

can

et

ithtedtly

dse

tn

coregy,

fore

idehose

(iii) If f :X → Y is continuous andA B ⊆ X, thenf (A) f (B).(iv) If A′ A ⊆ X andB ′ B ⊆ Y thenA′ × B ′ A × B in X × Y .(v) If W ⊆ X × Y is open andS T ⊆ X, then

y ∈ Y | ∀x ∈ T . (x, y) ∈ W ⊆ int

y ∈ Y | ∀x ∈ S . (x, y) ∈ W

.

Assertion (i) generalizes the fact that a closed subset of a compact space is comp(ii) the statement that a compact subset of a Hausdorff space is closed, (iii) the facontinuous maps preserve compactness, (iv) the Tychonoff theorem in the finite ca(v) the fact that ifX is compact then the projectionπ :X × Y → Y is a closed map, whicis one half of the Kuratowski–Mrova theorem. Concerning (v), notice that the projeis closed if and only if for every open setW ⊆ X × Y the setY \ π(X × Y \ W) is open,and that this set isy ∈ Y | ∀x ∈ X . (x, y) ∈ W . Hence the classical case occurs whenX

is compact andS = T = X. The proofs are modifications of the classical proofs andsafely be left to the reader.

Remark 4.4. It is well known and easy to verify that for a Hausdorff spaceX, thecompactly generated topology agrees with thek-topology, defined by declaring a substo bek-open if its intersection with each compact subspace is relatively open.

Theorem 4.5. For a Hausdorff spaceX the compactly generated topology agrees wthe core compactly generated topology(and hence with the locally compactly generatopology). In particular, X is compactly generated if and only if it is core compacgenerated.

Proof. Suppose thatU is open in the compactly generated topology, that is,U has theproperty that its inverse image is open for all probesi :K → X, whereK is a compactsubset ofX and i is the inclusion map. (This is equivalent toU ∩ K is open inK forall compact subspacesK.) Let p :C → X be a probe, whereC is core compact. We neeto showp−1(U) is open inC. If p(C) ∩ U = ∅, then this is clearly the case. Otherwilet x ∈ p−1(U). By core compactness pickV,W open inC such thatx ∈ V W C.By Lemma 4.3,p(V ) p(W) p(C) ⊆ X, thusp(V ) ⊆ p(W) X, and hence the seA = p(V ) is compact. ThereforeU ∩A is open inA. Sincep(V ) ⊆ A, there exists an opesetQ in C such thatx ∈ Q ⊆ V andp(Q) ⊆ U ∩ A ⊆ U . It follows thatp−1(U) is openin C, and hence thatU is open in the core compactly generated topology. Since thecompactly generated topology is always contained in the compactly generated topolowe are done.

Since the notion of compactly generated and core compactly generated collapseHausdorff spaces, it is natural to wonder if they coincide in general. That this is not thcase was shown by Isbell [20], who gave an example of a locally compactT1 (but clearlynon-Hausdorff) space that is not compactly generated.

However, the notions of core compactly generated and compactly generated do coincfor large classes ofT0-spaces. We give some examples, which include most of t


investigated in [15]. We begin with an elementary, but useful lemma. Recall that the

allroperty

orts

is

lypen,Scott-fty.

these

in

specialization order on the points of a topological space is defined byx y if and onlyx ∈ y. Since this is equivalent to saying that every neighbourhoodofx is a neighbourhoodof y, we see that open sets are upper sets in the specialization order.

Lemma 4.6.

(i) If the Sierpinski space(see example(2) after Lemma3.12) is C-generated then asubsetU of X which is open in theC-generated topology ofX is an upper set in thespecialization order ofX.

(ii) The Sierpinski space isC-generated if and only ifC contains a spaceC in which notevery open subset is closed.

(iii) The Sierpinski space is sequentially generated and hence compactly generated.

Proof. (4.6) Let U be aC-generated open set inX containingx and supposex y inthe specialization order. The mapp from S to X, sending 0 tox and 1 toy, is triviallycontinuous. BecauseS is C-generated,p−1U is open inS. But 0∈ p−1U , so 1∈ p−1U ,i.e.,y ∈ U as required.

(4.6) If U ⊆ C ∈ C is not closed, then the mapq :C → S that takes everyc ∈ U to 1 andall other points to 0 is a quotient, henceS is C-generated. Conversely, the property thatopen subsets are closed is preserved by disjoint sums and quotients. Hence, if this pis satisfied by all spaces inC then it is satisfied by allC-generated spaces.

(4.6) Immediate. The preceding lemma is quite useful, because in checking whether a certain space

class of spaces isC-generated, we needonly check that theC-generated open upper seare open in the original topology.

Theorem 4.7. Any directed-complete poset(dcpo) endowed with the Scott topologycompactly generated.

Proof. Let X be a dcpo under the Scott topology. By the preceding lemma, we need oncheck that an upper set inX that is open in the compactly generated topology is Scott-oor equivalently that a lower set that is closed in the compactly generated topology isclosed. LetF be such a lower set and letD be a directed subset ofF . If the supremum oD in not again inF , then we may assume thatD is a counterexample of least cardinaliBy the theorem of Iwomura (see, for example, [23]) one can writeD as a well-ordered (byinclusion) family of directed sets of lower cardinality. Then the supremum of each ofsets must belong toF , and these suprema form a well-ordered subsetC1 of F . Close upthis well-ordered chain under sups to obtain a chainC closed under sups inX that is alsowell-ordered. (LetA be a non-empty subset ofC. Pick a1 ∈ A, thena2 ∈ A with a2 < a1and continue this process. Either it terminates at the least element ofA or one obtains astrictly decreasing sequence inA. In the latter case, for eachai , we can choose abi ∈ C1such thatai+1 < bi ai . But the sequence(bi) is then an infinite decreasing sequenceC1, which violates the fact it is well-ordered.)


Now the chainC is a complete chain in its order, and hence is a compact Hausdorffus

esre thepace

at afromopent is ane

ingd

es

d for

ian

space in its order topology. Since it is sup-closed inX, the embedding map is continuointo X under the Scott topology, and thus a probe. Since the inverse image ofF containsC1, a dense subset ofC, we conclude thatC ⊆ F . But C contains the sup ofC1, which isthe supremum ofD by construction.

An important class of locally compact spacesis that of so-called stably compact spac(see, for example, [15, Chapter VI]). One characterization of these is that they atopological spaces with topologyconsisting of the open upper sets in a compact pos(again seeloc. cit.).

Theorem 4.8. Stably compact spaces are compactly generated.

Proof. Let X be a stably compact space. By Lemma 4.6 we need only verify thcompactly generated open set that is an upper set is open. Consider the probeXwith the Hausdorff topology making it a compact pospace to the topology of theupper sets. If the inverse of an upper set is open for this probe, then clearly the seopen upper set, and hence open in the stablycompact topology. We use here implicitly thfact that the order of specialization agrees with the partial order of the pospace.

One could envisage a theory of stably compactly generated spaces, but the precedtheorem shows that this theoryyields precisely the same spacesas the compactly generatespaces.

5. Products and function spaces

In this section we consider in more detail the topology of products and function spacin TopC , for productiveC. We show that finite products in TopC do not vary when weenlargeC. However, as we shall see, this invariance property fails for infinite limits anfunction spaces. Nevertheless, we shall obtain various useful characterizations of functionspaces in the different categories.

To begin with, letC be an arbitrary collection of topological spaces. ForC-generatedX,Y , we write X ×C Y for the binary product in the category TopC . Trivially, if thetopological productX × Y is itself C-generated thenX ×C Y = X × Y . Further, we sawin Section 3 that the topological productX × Y of two spacesX and Y is also theproduct in MapC by Lemma 3.9. Since the functorC : MapC → TopC is an equivalenceof categories, it follows that the categorical product of two spacesX andY in TopC isgiven byX ×C Y = C(X × Y ).

To proceed further, we assume henceforth thatC is productive. Thus TopC is Cartesianclosed, and we write[X ⇒C Y ] for exponentials in this category. By the proof of Cartesclosedness, and the equivalenceC : MapC → TopC , we have[X ⇒C Y ] = C(MapC[X,Y ])(n.b. forC-generated spaces, the set MapC(X,Y ) equals C(X,Y )).


Example 5.1. We show that finite products in TopC are, in general, strictly finer thanscreterly,

blece

ct

pty

e

lfe

is an

s as

topological products. Assume that the one-point compactification of a countable dispace isC-generated. ThusC may be any of our main examples, see Definition 3.3. Cleaevery sequential space isC-generated. LetNω be the Baire space, i.e., the countatopological product of the (discrete) non-negative integers. This is sequential, henC-generated. We show that the TopC product[Nω ⇒C N]×C N

ω does not have the produtopology.

By the Cartesian closedness of TopC , the set

P = (f, γ ) | f (γ ) = 0

is open (indeed clopen) in[Nω ⇒C N]×C Nω. We prove that this set contains no non-emsubsets of the formU × V , whereU ⊆ [Nω ⇒C N] andV ⊆ N

ω are open.Suppose, for contradiction, thatU ×V ⊆ P . Writing γ = γ0γ1γ2 · · · for elements ofNω

(i.e., infinite sequences), we can assume thatV has the form

V = γ ω ∈ N | (γ0, . . . , γk−1) = α

for somek ∈ N andα ∈ Nk , as such sets form a basis forNω. Let 2= 0,1 have thediscrete topology. Then 2ω is Cantor space, which is sequential, henceC-generated. Theris a continuous functionh : 2ω × Nω → N defined by

h(δ, γ ) =

0 if all of δ0, . . . , δγk are 0,1 otherwise.

Clearly, h : 2ω ×C Nω → N is also continuous (in fact 2ω × Nω is sequential so is itsethe product in TopC ). By the Cartesian closedness of TopC , the exponential transposh : 2ω → [Nω ⇒C N] is continuous. We claim that(h)−1U = 0ω (where 0ω is theconstant 0 sequence). This is a contradiction, because the singleton set0ω is not openin 2ω.

To verify the claim, supposeh(δ) ∈ U . Then, asU × V ⊆ P , for all γ with(γ0, . . . , γk−1) = α, it holds that(h(δ), γ ) ∈ P , i.e., h(δ)(γ ) = 0. That is,h(δ, γ ) = 0 forall γ with (γ0, . . . , γk−1) = α. So, by the definition ofh, indeedδ = 0ω.

A related example to the above appears as Proposition 7.3 of [17].

Definition 5.2 (Saturation). The saturationof a productiveC is the collectionC of C-generated spaces which are exponentiable in Top.

It is clear that saturation is an idempotent inflationary operation. The followingimmediate consequence of Lemmas 2.5 and 3.2.

Lemma 5.3.

(i) C is the largest class of exponentiable spaces that generates the same spaceC.Moreover, for any spaceX, the coreflectionsCX andCX coincide.

(ii) C is closed under arbitrary disjoint sums and finite topological products.(iii) A space isC-generated if and only if it is a quotient of a space inC.


Theorem 5.4.

p

l

ll

ents

ne

a,

m

s a.

our

ther to

he.e

le

eeis

(i) If A,B,Y are C-generated andq :A → B is a topological quotient, then the maq × idA :A×C Y → B ×C Y is a topological quotient.

(ii) If X andY areC-generated spaces withY exponentiable inTop, then the topologicaproductX × Y is C-generated.

(iii) If X andY areC-generated, thenX ×C Y = X ×E Y , whereE denotes the class of aexponentiable spaces.

Proof. A quick category-theoretic proof of (i) is to observe that topological quotibetweenC-generated spaces are just coequalizers in TopC , and that the functor(−)×C Y

preserves colimits, because it is a left adjointby Cartesian closedness. Alternatively, ocan easily adapt the proof of Lemma 2.6.

In order to prove (ii), we may assume thatC is saturated. By Lemma 5.3(iii) there isquotient mapq :C → X with C ∈ C. By Lemma 2.6 and the fact thatY is exponentiableq × idY :C × Y → X × Y is a quotient map. By the assumption of saturation,Y ∈ C,and henceC × Y ∈ C by Lemma 5.3(ii). The desired conclusion then follows froLemma 5.3(iii).

Regarding (iii), we may again assume thatC is saturated and hence that there iquotient mapq :C → X with C ∈ C. By (i), q × idC :C ×E Y → X ×E Y is a quotient mapBut, by (ii), C ×E Y = C ×Y = C ×C Y . HenceX ×E Y , being a quotient of aC-generatedspace, is itselfC-generated, and thus the product in the category ofC-generated spaces.

Of course, by instantiatingC appropriately, the above result specializes to allexample categories, as does the following useful consequence.

Corollary 5.5. If D ⊆ C is productive then the inclusion functorTopD → TopC preservesfinite products.

We now give examples showing that the above preservation result extends neiinfinite products nor to exponentials.

Example 5.6. Let D contain only the one-point space, so theD-generated spaces are tdiscrete spaces. As in Example 5.1, letC be such that every sequential space isC-generatedLet N be the discrete non-negative integers. Then,N

ω with the product topology is thBaire space, which is not discrete, but is sequential. Thus the countable powerNω in TopCis the Baire space, and the inclusion functor TopD → TopC does not preserve countabproducts.

Further,N is exponentiable in Top, and the exponential[N ⇒ N] is homeomorphic tothe Baire space, and hence coincides with the exponential[N ⇒C N] in TopC . Becausethe Baire space is not discrete, the inclusion functor TopD → TopC does not preservexponentials. The same example shows that the inclusion TopD → Top does not preservfunction spaces that exist in Top, i.e., those where the domain of the function spaceexponentiable.


Example 5.7. Let S contain only the one-point compactification ofN, so theS-generatedes,

wer

lcts.ialtial

onsiderthe

ctly

d.

mer

zernativelvesll

iated

spaces are the sequential spaces. LetC include, at least, all compact Hausdorff spacso every compactly-generated space isC-generated. Consider 2= 0,1 with the discretetopology. Letκ be any uncountable cardinal. By Tychonoff’s Theorem, the infinite po2κ in Top is a compact Hausdorff space, and henceC-generated. Thus 2κ is theκ-foldpower in TopC . Consider the set

γ ∈ 2κ | γi = 1 for uncountably manyi ∈ κ.

This is sequentially open, but not open in the product topology. Thus 2κ is not a sequentiaspace, and so the inclusion functor TopS → TopC does not preserve uncountable produ

Let X be a discrete space of cardinalityκ . ThenX is exponentiable and the exponent[X ⇒ 2] in Top is homeomorphic to 2κ , and hence coincides with the exponen[X ⇒C 2] in TopC . Thus[X ⇒C 2] is not sequential, and so the inclusion TopS → TopCdoes not preserve exponentials. The same example shows that the inclusion TopS → Topdoes not preserve exponentials between spaces whose exponential exists in Top.

Example 5.8. Let E be the collection of all exponentiable spaces. Thus a space isE-gene-rated if and only if it is core compactly generated. We show that the inclusion TopE → Topdoes not preserve exponentials between spaces whose exponential exists in Top. Cthe space[Nω ⇒E N], as in Example 5.1. The space 2 is trivially exponentiable, andexponential[2 ⇒ [Nω ⇒E N]] in Top is easily seen to be[Nω ⇒E N]2, i.e., [Nω ⇒EN] × [Nω ⇒E N] with the product topology. We show that this is not core compagenerated. BecauseN is a (continuous) retract ofNω, it holds that[N ⇒E N] is a retractof [Nω ⇒E N]. But [N ⇒E N] coincides with the exponential[N ⇒ N] in Top, which ishomeomorphic toNω, because the latter is (sequential hence) core compactly generateThus Nω is a retract of[Nω ⇒E N]. It follows that [Nω ⇒E N] × Nω is a retract of[Nω ⇒E N]× [Nω ⇒E N]. But if the latter were core compactly generated then the forwould be too (because retracts are quotients), which contradicts Example 5.1. Thus theexponential[2 ⇒E [Nω ⇒E N]] in TopE differs from the exponential[2 ⇒ [Nω ⇒E N]]in Top.

As function spaces vary across the different categories, it is interesting to characterithe topologies that arise in the different cases. To this end, we consider various altetopologies on the set C(X,Y ) of continuous maps. Although the topologies themsediffer, we shall prove that, for any productiveC, the associatedC-generated topologies acoincide with the TopC exponential[X ⇒C Y ].

The space MapC[X,Y ] defined in Section 3 is already one example whose assocC-generated topology is[X ⇒C Y ]. The definition of the topology on MapC[X,Y ] makesuse of the collectionC of generating spaces. We now consider topologies on C(X,Y )

determined instead byX andY alone.

Definition 5.9 (Isbell topology). The space[X ⇒I Y ] is the function space C(X,Y )

endowed with theIsbell topology, which has subbasic open sets

〈U,V 〉 = f | f −1(V ) ∈ U

,

whereV ⊆ Y is open andU is a Scott-open family of open subsets ofX.


The familiar compact-opentopology on C(X,Y ) is given by subbasic opens of the

veopeng1], an

sesr

r

thesey

gy.

-on-

d in

e

form f | f (K) ⊆ V , for compactK ⊆ X and openV ⊆ Y . When X is sober (inparticular if it is Hausdorff) it is a straightforward consequence of the Hofmann–MisloTheorem [15, Theorem II-1.21] that the Isbell topology coincides with the compact-topology whenever the Scott topology on the lattice of opens ofX has a basis consistinof filters. This latter property holds for allCech-complete spaces, see [10, Theorem 4.class that includes all complete metric spaces. In general, the Isbell topology is finer thathe compact-open topology.

Definition 5.10 (Natural topology). The space[X ⇒ Y ] is the function spaceC(X,Y )

endowed with thenatural topology, that is, the finest topology making all transpof :A → C(X,Y ) continuous wheneverf :A × X → Y is continuous, asA ranges oveall topological spaces.

A topology on C(X,Y ) is often called asplitting topology if f is continuous whenevef is. Hence the natural topologyis the finest splitting topology.

Proposition 5.11. If X is an exponentiable space, then:

[X ⇒ Y ] = [X ⇒I Y ] = [X ⇒ Y ],and this topology can also be given by subbasic open sets of the form

f | U f −1V,

generated by open setsU ⊆ X andV ⊆ Y .

The facts stated in the above proposition can be found in [11,18,29,30,13]. Inreferences, it is also shown that the Isbell topology is splitting. Thus the natural topologis always finer than the Isbell topology.

Example 5.12. The natural topology is, in general, strictly finer than the Isbell topoloLet Nω be the Baire space (see Example 5.1). Consider the set C(Nω,N). We claim thatthe set

F = f | f (

f(0ω

)f

(1ω

)f

(2ω

) · · ·) = 0

(wherenω is the constant sequence) is open in[Nω ⇒ N] but not in[Nω ⇒I N].Concerning the Isbell topology,Nω is complete-metrizable, hence, by [10, Theo

rem 4.1],[Nω ⇒I N] is the compact-open topology. It is not hard to show that no nempty finite intersection of subbasic sets of the compact-open topology is containeF

(cf. [17, Proposition 7.5]). ThusF is not open in[Nω ⇒I N].Regarding the natural topology, suppose thatg :A × N

ω → N is continuous. Then thfunctionh :A → N defined by

h(a) = g(a, g

(a, 0ω

)g(a, 1ω

)g(a, 2ω

) · · ·)is continuous. Soh−10 is open inA. But h−10 = (g)−1(F ), where the functiong :A → C(Nω,N) is the transpose ofg. Thus(g)−1(F ) is open, for every continuousg. It


follows that if F is added as an open set to[Nω ⇒ N] then the resulting topology is still

ology

se

oners

ap

ralge the

e on

lows

osee, let

basee

us, topt

splitting, hence equal to[Nω ⇒ N]. SoF is indeed open in[Nω ⇒ N].

We next develop an alternative characterization of the natural topology as the topof continuous convergence. A filterΦ on C(X,Y ) is said toconverge continuouslytof0 ∈ C(X,Y ) if for any filter G converging tob in X, the filter generated by the filter baF(G) | F ∈ Φ, G ∈ G converges tof0(b), where we writeF(G) for

⋃f (G) | f ∈ F .Thetopology of continuous convergenceis obtained in the standard fashion wheneveris given a family of convergent filters: a setU is open if whenever any of the given filteconverges to a point inU , thenU must belong to the filter.

A topology on C(X,Y ) is called conjoining if f :A × X → Y is continuouswheneverf :A → C(X,Y ) is. This is equivalent to the continuity of the evaluation mE : C(X,Y ) × X → Y . Notice that a topology on C(X,Y ) is exponential if and only if it isboth splitting and conjoining. Hence an exponential topology must agree with the natutopology since the natural topology is the finestsplitting topology and since any splittintopology is contained in any conjoining topology—see, e.g., [11] or [13]. We havfollowing additional observation.

Proposition 5.13. The natural topology and the topology of continuous convergencC(X,Y ) coincide and are the intersection of all conjoining topologies.

Proof. We first show that the topology of continuous convergence is splitting, and thenthat it is the intersection of a collection of conjoining topologies. The result then folfrom the fact that any splitting topology is contained in any conjoining topology.

It is splitting: Letf :A×X → Y be a continuous map. In order to show that its transpf :A → C(X,Y ) is continuous with respect to the topology of continuous convergencF be a filter converging toa0 ∈ A in the topology ofA. Let G be a filter converging tob ∈ X. Then the filter generated by the filter basef (F × G): F ∈ F , G ∈ G convergesto f (a0, b). Sinceb was arbitrary, this means that the filter generated by the filterf (a) | a ∈ S | S ∈ F continuously converges tof (a0), and hence converges in thtopology of continuous convergence.

It is the intersection of conjoining topologies: LetΦ be a filter converging tof0 ∈C(X,Y ) in the topology of continuous convergence, and consider the topologyOΦ,f0 onC(X,Y ) induced by the singleton family of convergence relationsΦ → f0. It is clear thatthe topology of continuous convergence is the intersection of all such topologies. Thconclude, it is enough to show that this topology on C(X,Y ) makes the evaluation maE : C(X,Y ) × X → Y continuous. Notice that the open sets ofOΦ,f0 are either sets thamiss f0 or sets that (containf0 and) belong to the filter. Assume thatE(f,x) = f (x)

belongs to an open setV ⊆ Y . If f = f0 then f is open, andf × f −1(V ) is anopen set carried intoV by evaluation. Otherwise, because the neighbourhood filter ofx

converges tox, the filter generated by the filter basef (x ′) | f ∈ S, x ′ ∈ T | S ∈ Φ,x ∈ intT ⊆ X converges tof0(x) by definition of continuous convergence. ThusE isindeed continuous.


Remark 5.14. In the Cartesian closed category of filter spaces (also called convergenceconver-er, by

main

e

ce

ynfore the

ologyletial7

that

spaces), see [17], the exponential convergence structure is given by continuousgence of filters. Thus, for topological spacesX,Y , the topology associated to the filtspace exponential on C(X,Y ) is the topology of continuous convergence and henceProposition 5.13, the natural topology. This fact is exploited in thesynthetic topologyof [12], see, in particular, Lemma 9.1 ofop. cit.

Having now thoroughly examined the Isbell and natural topologies, we state ourcharacterization of exponentials in TopC .

Theorem 5.15. For C-generated spacesX,Y , we have

[X ⇒C Y ] = C([X ⇒I Y ]) = C

([X ⇒ Y ]).Before proving the theorem we give an application, whereS is the Sierpinski space (se

example (2) after Lemma 3.12).

Corollary 5.16. If every compact Hausdorff space isC-generated then the function spa[X ⇒C S] has the Scott topology of the pointwise specialization order.

Proof. It is immediate from the definition of theIsbell topology that the Scott topologand Isbell topology agree on the function space[X ⇒C S]. But the pointwise specializatioorder is a complete lattice isomorphic to that of open sets and hence a dcpo. Thereresult follows from Theorems 5.15 and 4.7.

To prove Theorem 5.15, we develop an alternative characterization of the topMapC[X,Y ] used in the definition of[X ⇒C Y ]. The motivation is to obtain a simpdescription of a subbasis for MapC[X,Y ], close in spirit to the subbasis for exponentopologies given in Proposition 5.11. This characterization will prove useful in Sectionbelow.

Definition 5.17. The relation,C , on subsets of a topological spaceX is defined byS C T iff there exist aC-probep :C → X and open subsetsW V ⊆ C such thatS ⊆ p(W) andp(V ) ⊆ T .

Note that it is possible to haveC =D generating the same class of spaces but suchthe relationsC andD differ.

Lemma 5.18.

(i) S C T impliesS T .(ii) If S C T andU is an open cover ofT then there exist finitely manyS1 C U1, . . . ,

Sk C Uk , with eachUi ∈ U , such thatS ⊆ S1 ∪ · · · ∪ Sk .(iii) If S C T then there existsR such thatS C R C T .(iv) If f :X → Y is continuous andS C T thenf (S) C f (T ).


(v) If D ⊆ C thenS D T impliesS C T .

as 4.2

ges of

t

basice

of

(vi) If S C T in X if and only ifS C T in CX.

Proof. We just prove statement (ii), as the others are easy consequences of Lemmand 4.3. Suppose thatS C T andU is an open cover ofT . There exists a probep :C → X

with opensW V ⊆ C such thatS ⊆ p(W) andp(V ) ⊆ T . For eachc ∈ V , there existsUc ∈ U such thatx ∈ p−1Uc. Hence, by core compactness, there exists openWc ⊆ X suchthat c ∈ Wc p−1Uc. Then Wc | c ∈ V is an open cover ofV , hence it has a finitesubfamily Wc1, . . . ,Wck that coversW . For i with 1 i k, defineSi = p(Wci ) andUi = Uci . This clearly has the required properties.Definition 5.19 (C-topology). The space[X ⇒C Y ] is the function space C(X,Y )

endowed with theC-topologywhich has subbasic open sets

[S,V ]C = f | S C f −1(V )

,

whereS ⊆ X is arbitrary andV ⊆ Y is open.

Remark 5.20. Suppose thatC consists of locally compact spaces. Then theC -topologyreduces to the compact-open topology, where the compact sets run over the imacompact sets in the domains of probes from members ofC.

Proof. Suppose thatf ∈ [S,V ]C , i.e., S C f −1(V ). Then there existsA ∈ C, a probep :A → X, andT W ⊆ A such thatS ⊆ p(T ) andp(W) ⊆ f −1(V ). It follows thatT W ⊆ (fp)−1(V ). By local compactness we find for each point of(fp)−1(V ) acompact neighbourhood containing that point that is contained in the open set(fp)−1(V ).Then finitely many of these neighbourhoods coverT . If we denote their finite union byKthen we have thatK is compact andT ⊆ K ⊆ (fp)−1(V ). It follows that S ⊆ p(K),f carriesp(K) into V , hence thatp(K) C f −1(V ) (sinceK is compact), and thathe subbasic compact-open neighbourhood ofg | p(K) ⊆ g−1(V ) of f is contained in[S,V ]C . Hence the identity map from the compact-open topology to theC-topology iscontinuous. That it is continuous in the opposite direction is immediate, since subopens in the compact-open topology of the formg | p(K) ⊆ g−1(V ) are a special casof C-open sets. Proposition 5.21. For C-generated spacesX,Y , [X ⇒C Y ] = MapC[X,Y ].

Proof. To show that MapC[X,Y ] refines[X ⇒C Y ], we establish that[S,V ]C is open inMapC[X,Y ]. Considerf ∈ [S,V ]C . Then there exists a probep :C → X and openW W ′ ⊆ C such thatS ⊆ p(W) andp(W ′) ⊆ f −1V . The setG = g :C → Y | W g−1V is open in[C ⇒ Y ] (because[C ⇒ Y ] has the Isbell topology). So, by the definitionMapC[X,Y ] from Section 3, we have thatT −1

p G is open in MapC[X,Y ]. We show that

f ∈ T −1p G ⊆ [S,V ]C . As p(W ′) ⊆ f −1V , we haveW W ′ ⊆ (f p)−1V , so indeed

f ∈ T −1p G. Also, if h ∈ T −1

p G, thenW (h p)−1V , where(h p)−1V is open inC.

BecauseS ⊆ p(W) andp((h p)−1V ) ⊆ h−1V , we haveS C h−1V , i.e.,h ∈ [S,V ]C ,as required.


Conversely, to establish that[X ⇒C Y ] refines MapC[X,Y ], we show, for any probe,

ess

e

g the

ly

arationourerated

s

e

p :C → X, that the functionTp : [X ⇒C Y ] → [C ⇒ Y ] is continuous. Accordinglyconsider any subbasic openG = g | W g−1V of [C ⇒ Y ], given by opensW ⊆ C andV ⊆ Y , as in Proposition 5.11, and anyf ∈ T −1

p G, i.e.,W (f p)−1V . By interpolation,

there exists openW ′ ⊆ C with W W ′ (f p)−1V . Thenp(W ′) C p((f p)−1V ) ⊆f −1V , sof ∈ [p(W ′),V ]C . Also, for anyh ∈ [p(W ′),V ]C , we havep(W ′) ⊆ h−1V , soW W ′ ⊆ p−1(p(W ′)) ⊆ p−1(h−1V ) = (h p)−1V , and henceh ∈ T −1

p G. We have

shown thatf ∈ [p(W ′),V ]C ⊆ T −1p G. ThusT −1

p G is indeed open in[X ⇒C Y ]. Proof of Theorem 5.15. Let p :A → [X ⇒C Y ] be a probe. By the Cartesian closednof C-generated spaces, the function

(z, x) → p(z)(x) :A×C X → V

is continuous. Moreover, asA,X areC-generated spaces andA is core compact,A×C X

is just the topological productA × X. By definition of the natural topology, we havthatp :A → [X ⇒ Y ] is continuous. Since[X ⇒C Y ] is C-generated, it follows that itstopology refines the natural topology, which in turn refines the Isbell topology. TakinC-generated topologies, we conclude that[X ⇒C Y ] = C([X ⇒C Y ]) refinesC([X ⇒ Y ]),which refinesC([X ⇒I Y ]).

However,[X ⇒I Y ] refines[X ⇒C Y ] because, by Lemma 5.18(i), (iii), the famiU ⊆ X | U open andS C U is Scott open. Thus, by Proposition 5.21,[X ⇒I Y ] refinesMapC[X,Y ]. SoC([X ⇒I Y ]) refinesC(MapC[X,Y ]) = [X ⇒C Y ].

We close this section with some simple observations about how the standard sepaxioms may imposed upon the categories ofC-generated spaces. The theorem justifiescomment in Section 1 that our techniques implicitly cover the case of compactly genHausdorff spaces.

Theorem 5.22.

(i) Limits in TopC of diagrams ofT2 spaces are againT2 spaces.(ii) Disjoint sums inTopC of T2 spaces are againT2 spaces.(iii) If X,Y areC-generated spaces andY is T2 then also[X ⇒C Y ] is T2.

Moreover, analogous statements hold ifT2 is replaced throughout byT1 or T0.

Proof. For (i), we have that any categorical limit in Top ofT2 spaces is againT2. The limitin TopC is theC-generated topology, which is finer, hence againT2.

Statement (ii) is obvious.For (iii), we show that[X ⇒C Y ] is T2. Supposef = g ∈ C(X,Y ). Then there exist

x0 ∈ X with f (x0) = g(x0), hence there exist disjoint opensU,V ⊆ Y with f (x0) ∈ U andg0(x0) ∈ V . The setsF = h | h(x0) ∈ U andG = h | h(x0) ∈ V are easily seen to bdisjoint opens in[X ⇒C Y ] with f ∈ F andg ∈ G as required.

The proofs for theT1 andT0 cases are similar.


Note that the separation axioms are not preserved by quotients, hence the omission ofhisus theed

atof

and7, we

seents

ed to

quotients from the statement of the theorem. Note, further, that all spaces appearing in tsection, when giving examples of non-preservation of structure, were Hausdorff. Thsituation with respect to the preservation of products and function spaces is not improvby the imposition of separation axioms.

6. Quotients of countably based spaces

In this section we consider the full subcategory QCB of Top consisting of all spaces thare quotients of countably based spaces. Obviously, this is a subcategory of the categorysequential spaces, and hence a subcategory ofC-generated spaces, for anyC ⊇ S, where,as above,S = N∞. It is known that QCB is Cartesian closed, and that its productsfunction spaces are those in the category of sequential spaces [27,24]. In Sectionshall prove that, more generally, productsand function spaces in QCB coincide with thoof TopC , for everyC ⊇ S. To achieve this, we first obtain characterizations of quotiof countably based spaces based on variousnotions of “pseudobase”,loosely related tothe original such notion due to Michael [25]. For the purposes of this paper, we neconsider three notions of pseudobase.

Henceforth in this section, letC be productive withC ⊇ S. It follows that everysequential space isC-generated.

Definition 6.1 (Pseudobases). A collectionB of subsets of a topological spaceX is called

(i) an S-pseudobaseif given any convergent sequencexn → x and any open setUcontainingx, there existsB ∈ B such thatx ∈ B ⊆ U , andxn ∈ B for all sufficientlylargen,

(ii) a C-pseudobaseif, wheneverS C U ⊆ X with U open, there existsB ∈ B suchthatS ⊆ B ⊆ U ,

(iii) a -pseudobaseif, wheneverS U ⊆ X with U open, there existsB ∈ B such thatS ⊆ B ⊆ U .

The notion ofS-pseudobase is originally due to Schröder [28].

Lemma 6.2.

(i) AnyC -pseudobase is anS-pseudobase.(ii) Any-pseudobase is aC-pseudobase.(iii) Any topological base is anS-pseudobase and its closure under finite unions is a-

pseudobase(henceC-pseudobase).(iv) The closure under finite unions of anS-pseudobase is aS -pseudobase.

Lemma 6.3. If B is an S-pseudobase(respectively-pseudobase) for X, and Y ⊆ X

is any subset, then the familyB ∩ Y | B ∈ B is an S-pseudobase(respectively-pseudobase) for the relative topology onY .


The straightforward proofs of these lemmas are omitted.

order.

se.obasesnsist of

thatnsmerely

e havework

space

of

e

r

s

Recall that thesaturationof a subsetB of a topological space, denoted by↑B, is theintersection of its neighbourhoods, or, equivalently, its upper set in the specializationThe lower set ofB in the specialization order is denoted by↓B. If W is open, then↑B ⊆ W

if and only if B ⊆ W for any subsetB. It follows readily that if a collectionB is any of theabove types of pseudobase then the collection↑B | B ∈ B is the same type of pseudobaBecause of this, we shall henceforth assume, without loss of generality, that pseudalways consist of saturated sets. Also, all pseudobases we construct will indeed cosaturated sets.

The notions ofC - and -pseudobase share the slightly unfortunate propertytopological bases are not necessarily pseudobases (though their closures under finite unioare). This could be remedied by weakening these notions of pseudobase to requirethat there are finitely many members ofB, each contained inU , that coverS. However,as the weakened definitions are equivalent modulo closure under finite unions, wchosen to use the definition given above, because it is sometimes mildly simpler towith.

As the first result of this section, we show that every quotient of a countably basedhas a countable-pseudobase.

Proposition 6.4. Let ρ :X → Y be a quotient map, and assume thatX has a countablebaseB for the topology. The collection of finite unions of sets of the form↑ρ(B) withB ∈ B is a countable-pseudobase forY .

Proof. Let W be an open subset ofY and letA W . Let Bi : i = 1,2,3, . . . be anenumeration of allB ∈ B such thatρ(B) ⊆ W . SetAn = ⋃n

i=1 ↑ρ(Bi). We need to showthat someAn containsA. If not, choose for eachn someyn ∈ A \ An. Thenyn = ↓yn

missesAn = ↑An. The fact thatA W implies that the sequenceyn has a cluster pointyin W .

Fix somek and consider the setVk = ρ−1(W \⋃∞n=k ↓yn) = ρ−1(W)\⋃∞

n=k ρ−1(↓yn).Forx ∈ Vk , let On be a countable descending base atx. If eachOn meets the sequencesetsρ−1(↓yn) cofinally, then we can pick a sequencexnj → x such thatρ(xnj ) ∈ ↓ynj

for all nj . Pick B ∈ B such thatx ∈ B andρ(B) ⊆ W . Thenynj ∈ ↑ρ(xnj ) ⊆ ↑ρ(B) forall sufficiently largej , sincexnj → x. By the construction of theAn, we haveB ⊆ An

for all sufficiently large indicesn. It follows thatynj ∈ Anj for sufficiently largej . Thiscontradicts the construction of the sequenceyn. We conclude that for anyVk and anyx ∈ Vk, there exists some open set aroundx that misses all but finitely many of thρ−1(↓yn).

From the definition ofVk, if x ∈ Vk, thenx /∈ ρ−1(↓yn) for anyn k. Since the latteis a closed set and since from the preceding paragraph there is an open setU ⊆ ρ−1(W)

aroundx that hits at most finitely many of them, we can intersectU with the complementof all ρ−1(↓yn), n k, thatU hits, and obtain an open set containingx and contained inVk. It follows thatVk is open, and thusW \ ⋃∞

n=k ↓yn is open, sinceρ is a quotient map.Now it cannot be the case thaty ∈ W \⋃∞

n=k ↓yn, sinceyn clusters toy and eventuallymisses the open setW \ ⋃∞

n=k ↓yn. Thus it must be the case thaty ∈ ↓yn for somen k. Sincek was arbitrary we conclude thaty ∈ ↓yn for infinitely many indicesn. Let


p ∈ X such thatρ(p) = y ∈ W . There existsB ∈ B such thatp ∈ B, ρ(B) ⊆ W . Then

e

that

en,

r any

y

ur proofard)

pose

from all large enough integersn, we have↑ρ(B) ⊆ An. But for infinitely many indicesyn ∈ ↑y ⊆ ↑ρ(B). This contradictsyn /∈ An for all n. This contradiction completes thproof. Proposition 6.5. LetX be a topological space with a countableC-pseudobase. ThenCX

is the sequential topology onX.

Proof. SinceS ⊆ C, it holds thatCX refines the sequential topology. We must showeach sequential open setU is open inCX. If it is not, then there exists a probep :Q → X,whereQ ∈ C, such thatp−1(U) is not open inQ. Hence there existsy ∈ Q such thaty ∈ p−1(U), butV is not a subset ofp−1(U) for every open subsetV containingy. Thatis, p(V ) is not contained inU , for every open subsetV containingy.

Let B be a countableC-pseudobase forX. Let A consist of allB ∈ B such thatp(V ) ⊆ B for someV open containingy. BecauseQ is core compact, there exists opV ⊆ Q with y ∈ V Q, sop(V ) C p(Q) ⊆ X, whence, becauseB is aC -pseudobasethere existsB ∈ B such thatp(V ) ⊆ B. This shows thatA is non-empty. LetBn | n 1be an enumeration ofA. For eachn, we choosexn ∈ ⋂n

i=1 Bn such thatxn /∈ U (this ispossible sinceBi contains somep(Vi), andp(

⋂ni=1 Vi) ⊆ ⋂n

i=1 Bi , butp(⋂n

i=1 Vi) is notcontained inU ).

LetW be any open set containingp(y). Thenp−1(W) is open inQ and containsy. Thusby core compactness there exists some open setV containingy such thatV p−1(W).Thenp(V ) C p(p−1(W)) ⊆ W . SinceB is aC -pseudobase, there exists someB ∈ Bsuch thatp(V ) ⊆ B ⊆ W . ThenB ∈ A, i.e.,B = Bj for somej , and thusxn ∈ Bj ⊆ W forall n j . SinceW was an arbitrary open set containingp(y), we conclude thatxn → p(y).SinceU is a sequentially open set containingp(y), we thus obtain thatxn ∈ U for somen.But this contradicts the choice ofxn. Corollary 6.6. The sequential and core compactly generated topologies agree fospace with countable-pseudobase.

Proof. Any countable-pseudobase onX is also aE -pseudobase, whereE is thecollection of all core compact spaces. By Proposition 6.5,EX is the sequential topologonX. But EX is the core compactly generated topology onX.

Proposition 6.8 below, which shows thatany sequential space with countableS-pseudobase is a quotient of a countably based space, is due to Schröder, and ois adapted from [28,2]. We first recall, without proof, a standard (and straightforwtopological lemma.

Lemma 6.7. Let f :Y → X be a continuous function between sequential spaces. Supthat a sequence(xi) → x in X implies that there exist(yi) andy in Y such thatf (yi) = xi ,f (y) = x and (yi) → y in Y , i.e., convergent sequences lift. Thenf is a topologicalquotient map.


Proposition 6.8. Let X be a topological space with countableS-pseudobase. ThenX

yy.at the

base

tial,

f

equipped with the sequential topology is the quotient of a countably based space.

Proof. We assume that the elements of theS-pseudobaseB are enumerated asBn forn ∈ N.

We construct a countably based spaceR as follows. The elements ofR are pairs(x,β),wherex ∈ X andβ ∈ ∏

n∈N0,1 satisfy

(i) for all n ∈ N, x ∈ Bn if β(n) = 1, and(ii) for all neighbourhoodsV of x there existsn ∈ N such thatβ(n) = 1 andBn ⊆ V .

We topologizeΩ = ∏n0,1 with the product of the Sierpinski topologies (with1 the

only non-trivial open set). The topology ofX ×Ω is the product of the indiscrete topologonX with the topology just defined onΩ , and the topology ofR is the subspace topologThe topology onR has an alternative description as the coarsest topology such thsecond projectionR → Ω is continuous.

The topology ofR is countably based since that ofX × Ω is. Let p :R → X be thefirst projection function, where the imageX has its original topology. Let(x,β) ∈ R withp(x,β) = x ∈ U , an open subset ofX. Then there existsn ∈ N such thatβ(n) = 1 andx ∈ Bn ⊆ U . Consider the open setV of all (y, γ ) such thatγ (n) = 1. Theny ∈ Bn, sop(y, γ ) ∈ U . Thusp is continuous. It is easily verified from the properties of a pseudothatp is surjective.

Claim. For p :R → X, convergent sequences lift.

Suppose(xi) → x in X. Definey = (x,β) whereβ(n) = 1 if and only if x ∈ Bn and(xi) is eventually inBn. This is a well-defined element ofR because(xi) → x andB is anS-pseudobase. Defineyi = (xi, βi) whereβi(n) = 1 if and only ifxi ∈ Bn. We just need toshow that(yi) → y. Letβ(n) = 1. We must show that there existsk such that, for alli k,βi(n) = 1. But this follows from the definition ofβ andβi . This completes the claim.

Putting the above together, we have a second-countable spaceR and continuousp :R →X satisfying the hypotheses of Lemma 6.7. AsR is second-countable, hence sequenwe can factorp :R → X asR → Seq(X) → X. By Lemma 6.7 the mapR → Seq(X) is atopological quotient. Proposition 6.9. Let X be a topological space with countableC-pseudobaseB. ThenCX is a quotient of a countably based space and the closure ofB under finite unions andintersections is a countable-pseudobase forCX.

Proof. By Proposition 6.5,CX is the sequential topology onX. Moreover,B is a countableS-pseudobase onX. Thus the proof of Proposition 6.8 exhibitsCX as a quotient oR ⊆ X × Ω .

Let J be a finite subset ofN. Then the set(x,β) | β(n) = 1 for all n ∈ J


is a basic open set inX × Ω . Any point (x,β) ∈ R that is also in this basic open sett

asis

reand

hen it

uotientossibleed and

eteed

faset of

must be in⋂

n∈J Bn and conversely for any point inx ∈ ⋂n∈J Bn, one can construc

γ ∈ Ω such that(x, γ ) ∈ R. It follows that the images of the specified countable bof R consist of finite intersections of members ofB. Thus, by Proposition 6.4, theclosure under finite unions of saturates of finite intersections of members ofB is a -pseudobase for the topologyCX. But the members ofB, and hence their intersections, aalready saturated. Thus the-pseudobase is simply the closure under finite unionsintersections ofB.

The next theorem follows directly from our preceding results. Recall thatC is assumedto be productive withC ⊇ S, whereS = N∞.

Theorem 6.10. The following are equivalent for any spaceX:

(i) X is a quotient of a countably based space.(ii) X is a sequential space with a countableS-pseudobase.(iii) X is aC-generated space with a countableC-pseudobase.(iv) X is a core compactly generated space with a countable-pseudobase.

The equivalence of statements (1) and (2) is originally due to Schröder [27].

Corollary 6.11. If a core compact space is a quotient of a countably based space, tis itself countably based.

Proof. Take any core compact spaceX with countable-pseudobaseB. It is easy to seethat the collection of interiors of members ofB is a countable base forX.

Any quotient of a countably based space is core compactly generated, hence a qof a core compact space. The remark below shows, however, that it is not always pto exhibit a QCB space as a quotient of a space that is simultaneously countably bascore compact.

Remark 6.12. The Baire spaceNω (the product of countably many copies of a discrcountable space), although itself countably based, is not a quotient of any countably bascore compact space.

Proof. Suppose for contradiction there were a quotient mapq :X → Nω with X acountably based core compact space. ThenX has a countable baseUi of opensUi suchthatUi X. By Lemma 4.3(ii), the closures of the setsq(Ui) form a countable family ocompact sets, and it clearly coversNω. By the Baire category theorem, one of them hnon-empty interior. But then this interior is a non-empty locally compact open subsNω, andNω has no locally compact non-empty opens.


7. Countable limits and function spaces in QCB

gbled inFirstly,nd is

lusionts ofitsmbient

t such

oductquence

ndpoint

e

in

In this section, we investigate the behaviour of QCB spaces within any containincategory of C-generated spaces. The main resultstates that whenever a counta(categorical) limit of QCB spaces or a function space between QCB spaces is calculatethe category ofC-generated spaces then the resulting space is again a QCB space.this provides a self-contained proof that the category QCB has countable limits aCartesian closed (see [27,24] for alternative proofs). Secondly, it follows that the incof QCB in TopC preserves function spaces and countable limits. Thus, for quotiencountably based spaces, thetopologies underlying function spaces and countable limare, in a strong sense, canonical, for they are always the same irrespective of which acategory TopC they are calculated within. In contrast, Example 5.7 demonstrates thaa situation does not hold for more general spaces.

We briefly review the construction of countable limits in the category TopC . As is wellknown, any countable limit can be constructed from a combination of a countable prand an equalizer, so it suffices to describe the construction of these. Given a se(Xi)i0 of C-generated spaces, the countable product,

∏Ci Xi , in the category TopC is

defined by:∏i

CXi = C

(∏i

Xi

),

where the right-hand side is theC-generated topology of the topological product. IfX andY areC-generated spaces andf,g :X → Y are continuous, then the equalizerE in TopC isdefined by

EC = C(

x ∈ X | f (x) = g(x))

,

where the subset ofX on the right is given the subspace topology.

Proposition 7.1. The following conditions are equivalent forC:

(i) EveryQCB-space isC-generated.(ii) The one point compactification ofN is C-generated.(iii) Every sequential space isC-generated.

Proof. The one point compactification ofN is countably based, hence in QCB, aevery QCB space is sequential. Finally, we have already seen that if the onecompactification ofN is C-generated then so is every sequential space.

Henceforth in this section, we assume thatC is productive and satisfies any of thequivalent conditions of Proposition7.1.

Theorem 7.2.

(i) If f,g :X → Y are continuous maps betweenQCB spaces then their equalizerTopC is also aQCB space.


(ii) If (Xi)i0 is a sequence ofQCB spaces, then∏C

i Xi is a QCB space.

n the

the

ture

tenceence

use

es.

gy

otions

rivially,as a

e

talk

(iii) If X andY are QCB spaces, then so is[X ⇒C Y ].

Before proving the theorem, we state a corollary, which follows immediately, givefact that TopC is Cartesian closed and has arbitrary limits.

Corollary 7.3. The categoryQCB is Cartesian closed with countable limits. Moreover,inclusion ofQCB in TopC preserves function spaces and countable limits.

Although the first half of this corollary is already known [27,24], the result on strucpreservation is new, and is one of the main contributions of the present paper.

Remark 7.4. The results above say nothing about categorical colimits. In fact, the exisof countable colimits in QCB is straightforward, they are computed as in Top, and htrivially preserved by the inclusion of QCB in TopC .

To prove Theorem 7.2 we assume henceforth thatC is productive andC ⊇ S, as in theprevious section. The latter assumptioncan be made without loss of generality, becaTheorem 7.2 does not depend upon the choice of generating spaces for TopC .

Proof of Theorem 7.2(i). Let f,g :X → Y be continuous maps between QCB spacDefine E = x ∈ X | f (x) = g(x). By Lemma 6.3, a countable-pseudobase forXrestricts to a countable-pseudobase (henceC -pseudobase) for the relative topoloonE. Thus, by Proposition 6.9, the equalizerEC = C(E) is a QCB space.

To prove the remaining statements of Theorem 7.2, it is convenient to develop nof pseudosubbase. Given a familyB of subsets of a setX, we write

⋃(B) for the closure of

B under finite unions,⋂

(B) for the closure ofB under finite intersections, and⋃−⋂

(B)

for the closure ofB under finite intersections and unions.

Definition 7.5 (Pseudosubbases). A family B of saturated sets is said to be a-pseudo-subbase(respectivelyC -pseudosubbase) for a topological spaceX if

⋃−⋂(B) is a-

pseudobase (respectivelyC -pseudobase) forX.

The results on pseudobases from Section 6 adapt easily to pseudosubbases. Tany -pseudobase is aC-pseudosubbase. By Theorem 6.10, any QCB space hcountable-pseudobase. By Proposition 6.9, if a topological spaceX has a countablC -pseudo(sub)baseB, thenCX is a QCB space andB is a -pseudosubbase forCX.Hence a familyB is aC -pseudosubbase for aC-generated space,X, if and only if it isa -pseudosubbase for the space. Thus, forC-generated spaces, we shall henceforthunambiguously about pseudosubbases, without qualifying them as being of theC or variety.

By the above remarks, statement (ii) ofTheorem 7.2 follows immediately fromProposition 7.6 below, using the construction of countable products in TopC . Statement (iii)follows from Proposition 7.7, because[X ⇒C Y ] = C[X ⇒C Y ] by Proposition 5.21.


Proposition 7.6. Let (Xi)i0 be a sequence of topological spaces, where eachXi has

es

r.8,

es)

C -pseudosubbaseBi . Then∏

i Xi hasC -pseudosubbase:(B0 × · · · × Bn−1 ×

∏in

Xi

)| n 0, B0 ∈ B0, . . . ,Bn−1 ∈ Bn−1

,

which is countable if eachBi is.

Proposition 7.7. If X andY are C-generated spaces, with countable pseudosubbasAandB, respectively, then the set

((A,B)) | A ∈⋂

(A), B ∈⋃

(B),

where

((A,B)) = f ∈ C(X,Y ) | f (A) ⊆ B

,

is a countableC -pseudosubbase forX ⇒C Y .

The remainder of this section is devotedto the proofs of Propositions 7.6 and 7.7.

Lemma 7.8. Suppose thatW is a subbase for a topological spaceX. Then a familyB ofsaturated subsets ofX is aC -pseudosubbase forX if and only if, for everyS C W ∈W ,there existsB ∈ ⋃−⋂

(B) such thatS ⊆ B ⊆ W .

Proof. The only-if direction is trivial. For the if direction, suppose thatB satisfies thecondition stated. We must show it to be aC -pseudosubbase. Suppose then thatS C U

whereU ⊆ X is open. We have thatU = ⋃V , for some familyV consisting of finite

intersections of elements ofW . By Lemma 5.18(ii), for somek 0 there existS1 CV1, . . . , Sk C Vk with V1, . . . , Vk ∈ V and S ⊆ S1 ∪ · · · ∪ Sk . Then, for eachVi , wehaveVi = Wi1 ∩ · · · ∩ Wiki , with eachWij ∈ W , so Si C Wi1, . . . , Si C Wiki . Byassumption, for eachWij , there existsBij ∈ ⋃−⋂

(B) with Si ⊆ Bij ⊆ Wij . Thus, definingBi = Bi1 ∩ · · · ∩ Biki , we haveSi ⊆ Bi ⊆ Vi ; and, definingB = B1 ∪ · · · ∪ Bk , we haveS ⊆ S1 ∪ · · · ∪ Sk ⊆ B ⊆ V1 ∪ · · · ∪ Vk ⊆ U . Thus we indeed haveB ∈ ⋃−⋂

(B) withS ⊆ B ⊆ U . Proof of Proposition 7.6. Consider any basic open setU0 × · · ·× Un−1 × ∏

in Xi in theproduct topology

∏i Xi , given by opensU0 ⊆ X0, . . . ,Un−1 ⊆ Xn−1. Suppose thatS C

(U0 × · · · × Un−1 × ∏in Xi) where, by Lemma 5.18(iv), for eachj with 0 j n − 1,

we haveπj (S) C Uj , whereπj is the projection. Thus, there existsAj ∈ ⋃−⋂(Bj )

such thatπj (S) ⊆ Aj ⊆ Uj . Then it holds thatS ⊆ (A0 × · · · × An−1 × ∏in Xi) ⊆

(U0 × · · · × Un−1 × ∏in Xi), andA0 × · · · × An−1 × ∏

in Xi is in the closure undefinite intersections and unions of the candidateC -pseudosubbase. Thus, by Lemma 7this is indeed aC -pseudosubbase. The countability claim is obvious.

The proof of Proposition 7.7 requires a sequence of lemmas. The first adapts (and usLemma 4.3(iv), (v) to the×C product and theC relation.


Lemma 7.9. LetC be any collection of exponentiable spaces.

at if

(i) If X,Y areC-generated spaces,A′ C A ⊆ X andB ′ B ⊆ Y thenA′×B ′ A×B

in X ×C Y .(ii) If X,Y areC-generated spaces,W ⊆ X ×C Y is open andS C T ⊆ X then

y ∈ Y | ∀x ∈ T . (x, y) ∈ W ⊆ int

y ∈ Y | ∀x ∈ S . (x, y) ∈ W

.

Proof. (i) SupposeA′ C A ⊆ X andB ′ B ⊆ Y . Thus there existQ ∈ C, continuousp :Q → X andW ′ W ⊆ Q with A′ ⊆ p(W ′) andp(W) ⊆ A. By Lemma 4.3(iv), wehaveW ′ × B ′ W × B ⊆ Q × Y . Then, applying the continuous function

Q × Y = Q×C Yp ×C id−→ X ×C Y,

we obtain thatp(W ′) × B ′ p(W) × B in X ×C Y . But A′ × B ′ ⊆ p(W ′) × B ′ andp(W) × B ⊆ A × B, so indeedA′ × B ′ A × B in X ×C Y .

(ii) SupposeS′ C T ⊆ X and B ′ B ⊆ Y . Then here existQ ∈ C, continuousp :Q → X and U V ⊆ Q with S ⊆ p(U) and p(V ) ⊆ T . Applying the continuousfunction

Q × Y = Q×C Yp ×C id−→ X ×C Y,

we have, by Lemma 4.3(v),y ∈ Y | ∀q ∈ V .

(p(q), y

) ∈ W ⊆ int

y ∈ Y | ∀q ∈ U .

(p(q), y

) ∈ W.

But alsoy ∈ Y | ∀x ∈ T . (x, y) ∈ W

⊆ y ∈ Y | ∀q ∈ V .

(p(q), y

) ∈ W,

asp(V ) ⊆ T , andy ∈ Y | ∀q ∈ U .

(p(q), y

) ∈ W ⊆

y ∈ Y | ∀x ∈ S . (x, y) ∈ W,

asS ⊆ p(U), which proves the claim. Remark 7.10. For C closed under binary product, a similar argument shows thA′ C A ⊆ X andB ′ C B ⊆ Y thenA′ × B ′ C A × B in X ×C Y .

Lemma 7.11. Suppose thatX,Y are C-generated spaces, thatA is a countableC -pseudobase forX, that B is a countable-pseudobase forY , and thatF C [T ,V ]Cin X ⇒C Y , whereV ⊆ Y is open. Then, for anyS C T ⊆ X, there existA ∈ ⋂

(A)

andB ∈ B such thatF ⊆ ((A,B)) ⊆ [S,V ]C .

Proof. By Lemma 5.18(iii), there existsG with F C G C [T ,V ]C in X ⇒C Y . ByLemma 5.18(vi), we also haveF C G C [T ,V ]C in X ⇒C Y = C(X ⇒C Y ). By thecontinuity of evaluation

(X ⇒C Y ) ×C X → Y, (7.1)


the set(f, x) | f (x) ∈ V is an open subset of(X ⇒C Y ) ×C X. Thus, by Lemma 7.9(ii),

n

e

r

we obtain the second inclusion of

T ⊆ x | ∀f ∈ [T ,V ]C . f (x) ∈ V

⊆ intx | ∀f ∈ G . f (x) ∈ V

,

where the first inclusion is immediate from the definition of[T ,V ]C .Now take anyS C T in X. By Lemma 5.18(iii),S C S′ C T for someS′ ⊆ X.

Combining with the above inequalities, we obtain

S′ C intx | ∀f ∈ G . f (x) ∈ V

. (7.2)

Let Aii0 be an enumeration of the setA ∈ A | S′ ⊆ A ⊆ int

x | ∀f ∈ G . f (x) ∈ V

,

which is non-empty becauseA is aC-pseudobase forX. Let Bii0 be an enumeratioof the setB ∈ B | B ⊆ V , which is non-empty becauseB is a-pseudobase forY . Weclaim that there existsk 0 with F ⊆ ((A1 ∩ · · · ∩ Ak,Bk)).

To establish the claim, suppose, for contradiction, that it fails. Then, for eachi 0,there existsxi ∈ A1 ∩ · · · ∩ Ai andfi ∈ F such thatfi(xi) /∈ Bi . We first show that

S′ ∪ xi | i 0 intx | ∀f ∈ G . f (x) ∈ V

. (7.3)

For this, suppose thatU is an open cover of intx | ∀f ∈ G. f (x) ∈ V . Then, by (7.2) andLemma 5.18(i), (iii), there exists finiteU ′ ⊆ U with S′ C

⋃U ′. AsA is aC-pseudobas

for X, there existsAi such thatS′ ⊆ Ai ⊆ ⋃U ′. But, for j i, we havexj ∈ Ai . So

U ′ is a finite cover ofS′ ∪ xi | i j . Thus, by choosing opensU1, . . . ,Ui−1 ∈ U withx1 ∈ U1, . . . ,xi−1 ∈ Ui−1, we have thatU ′ ∪ U1, . . . ,Ui−1 is the required finite subcoveof S′ ∪ xi | i 0. This establishes (7.3).

As F C G in X ⇒C Y , we have, by (7.3) and Lemma 7.9(i),

F ×C(S′ ∪ xi | i 0) G×C int

x | ∀f ∈ G . f (x) ∈ V

in (X ⇒C Y )×C X. By the continuity of evaluation (7.1), it follows that

F(S′ ∪ xi | i 0) G

(int

x | ∀f ∈ G . f (x) ∈ V

) ⊆ V

in Y . Therefore, there existsB ∈ B with F(S′ ∪ xi | i 0) ⊆ B ⊆ V , becauseB is a-pseudobase forY . But thenB = Bi for somei, and alsofi(xi) ∈ Bi , contradicting thechoice offi andxi . This contradiction establishes that the claim is indeed true.

We have proved that there existsk with F ⊆ ((A1 ∩ · · · ∩ Ak,Bk)). It remains toverify that ((A1 ∩ · · · ∩ Ak,Bk)) ⊆ [S,V ]C . But, for f ∈ ((A1 ∩ · · · ∩ Ak,Bk)), we havef (A1 ∩ · · · ∩ Ak) ⊆ Bk ⊆ V . Therefore,

S C S′ ⊆ A1 ∩ · · · ∩ Ak ⊆ f −1(V ),

i.e.,f ∈ [S,V ]C as required. Proof of Proposition 7.7. Suppose thatX andY areC-generated spaces,A is a countablepseudosubbase forX andB is a countable pseudosubbase forY . Thus

⋃−⋂(A) is a-

pseudobase (henceC-pseudobase) forX, and⋃−⋂

(B) is a-pseudobase forY . Wemust show that the set

F =((A,B)) | A ∈

⋂(A), B ∈

⋃(B)


is a pseudosubbase ofX ⇒C Y .t

f

,

d in

ntducts,

whose

alults

msto thes

edithd, is

section.ls overve

Suppose thatF C [S,V ]C in X ⇒C Y , whereV ⊆ Y is open. By Lemma 7.8, isuffices to show that there existsG ∈ ⋃−⋂

(F) with F ⊆ G ⊆ [S,V ]C .For everyf ∈ [S,V ]C , we haveS C f −1(V ), so, by Lemma 5.18(iii), there existsTf

such thatS C Tf C f −1(V ). Thus the family[Tf ,V ]Cf∈[S,V ]C is an open cover o[S,V ]C . As F C [S,V ]C , there exist, by Lemma 5.18(ii),F1 C [T1,V ]C, . . . ,Fk C[Tk,V ]C , for someT1, . . . , Tk ∈ Tf | f ∈ [S,V ]C, with F ⊆ F1 ∪ · · · ∪ Fk .

For eachi with 1 i k, we haveFi C [Ti,V ]C andS C Ti . So, by Lemma 7.11there existAi ∈ ⋂

(⋃−⋂

(A)) = ⋃−⋂(A) andBi ∈ ⋃

(⋃−⋂

(B)) = ⋃−⋂(B) such

thatFi ⊆ ((Ai,Bi)) ⊆ [S,V ]C . Thus

F ⊆ F1 ∪ · · · ∪ Fk ⊆ ((A1,B1)) ∪ · · · ∪ ((Ak,Bk)) ⊆ [S,V ]C .

DefineG = ((A1,B1)) ∪ · · · ∪ ((Ak,Bk)). It remains only to verify thatG ∈ ⋃−⋂(F).

We show that each((Ai,Bi)) ∈ ⋂(F). For this, we haveAi = Ai1 ∪ · · · ∪ Aiki , with

eachAij ∈ ⋂(A), andBi = Bi1 ∩ · · · ∩ Bili , with eachBij ∈ ⋃

(B). Thus

((Ai,Bi)) =⋂

1jki

⋂1j ′li

((Aij ,Bij ′ )),

where each((Aij ,Bij ′ )) ∈ F . So indeed((Ai,Bi)) ∈ ⋂(F).

Remark 7.12. The results of this section show that, whenever QCB is containeTopC , then the inclusion preserves countablelimits and function spaces. This situationis not mimicked by the inclusion QCB→ Top. Given the results of the presesection, Example 5.1 shows that the latter inclusion fails to preserve finite proand Example 5.8 shows that it does not preserve exponentials between spacesexponential exists in Top.

Remark 7.13. In the case of two countably based spacesX,Y , the exponential[X ⇒C Y ](for productiveC satisfying any of the conditions of Proposition 7.1) is given by the naturtopology[X ⇒ Y ]. This is established by the following chain of reasoning. By the resof the present section,[X ⇒C Y ] coincides with the exponential in QCB. By [24, Theore2 & 3], the latter exponential is obtained as the topology associated (via quotient)exponential in Scott’s category of equilogicalspaces [3]. Finally, by [12, Appendix A], thiassociated topology is indeed the natural topology.

Remark 7.14. In [26, §3.3] a proof is given that, for the “type hierarchy” of iteratexponentials over the discrete natural numbers, the sequential topology coincides wthe compactly generated topology. This theorem, which is originally due to Hylanan immediate consequence of the structure preservation results of the presentFurthermore, our results establish analogous coincidences for iterated exponentiaarbitrary QCB spaces, and with respect to takingC-generated topologies for any productiC satisfying the conditions of Proposition 7.1.


8. Compact generation, a dual Hofmann–Mislove Theorem, and Lawson duality

pect toe usegy,

nsider.tice ofhich isa duale ofding ane ofters ofhat

act

ets is

from

atmpact

and

The compact saturated subsets of a topological space form a semilattice with resthe operation of union. In this section we investigate the following problem: can onthis semilattice of subsets to determine internally the core compactly generated topoloas opposed to the external determination via probes?

There are two additional related questions of a more technical nature that we coRecall that the Hofmann–Mislove Theorem asserts that a Scott-open filter in the latopen sets of a sober space is the open neighbourhood filter of its intersection, wa compact saturated set (see, for example, [15, Theorem II-1.21]). We considerproblem: under what conditions is it true that a Scott-open filter in the semilatticcompact saturated sets (ordered by reverse inclusion) is the filter of compact saturatesubsets of its union, which is an open set? One may view this question as providalternative approach to the theory of compactly generated spaces, since in the presencan affirmative answer the topology of open sets is determined by the Scott-open filthe semilattice of compact saturated sets. Finally we consider the problem of under wconditions the lattice of open sets of a topological space and its semilattice of compsaturated sets stand in Lawson duality to each other.

We shall have need of the following useful lemma. Recall that a family of subsfiltered if it forms a filter base.

Lemma 8.1. Let f :X → Y be continuous, whereX is a sober space. IfL is theintersection of a filtered familyK of compact saturated subsets ofX, then

⋂↑f (K) |K ∈ K = ↑f (L).

Proof. Clearly↑f (L) ⊆ ⋂↑f (K) | K ∈ K. Conversely suppose thaty /∈ ↑f (L). Then↓y ∩ f (L) = y ∩ f (L) = ∅. ThusL misses the closed setf −1(↓y). By the machineryof the Hofmann–Mislove Theorem [15, Theorem II-1.21] there existsK ∈ K such thatK ⊆ X \ f −1(↓y) and thusy /∈ ↑f (K). Lemma 8.2. Let X be a sober space. Then the topology determined by the probeslocally compact sober spaces agrees with the core compactly generated topology.

Proof. Let p :C → X be a probe from a core compactly generated spaceC. Since thespaceX is sober, the mapp factors through the sobrificationCs of C, which is a locallycompact space (see [15, Exercise V-4.9 and Proposition V-5.10]). Letps :Cs → X be thecorresponding probe from the sobrification. Then forW ⊆ X, p−1(W) if open inC if andonly if (ps)−1(W) is open inCs (Proposition V-5.10 of [15] again). It thus follows ththe core compactly generated topology is determined by all probes from locally cosober spaces intoX.

The next lemma establishes a connection between core compactly generated spacesfilters of compact saturated sets.


Lemma 8.3. If X is a sober space andK is a Scott-open filter of compact saturated sets,

a

t

he set

t

ly

ated

f theubsetsthis

tly

ruratedlter of

the

ge

then the setV = ⋃K is open in the core compactly generated topology.

Proof. We must show for a probep :C → X thatp−1(V ) is open. By the preceding lemmwe may restrict our attention to probesp :C → X whereC is locally compact sober.

Let F be the collection of all compact saturated subsetsF of C such that↑p(F) ∈ K.We claim thatF is a Scott-open filter of compact saturated subsets. SinceK is a filter, itfollows directly thatF is also a filter. Suppose thatM is a filtered family of compacsaturated subsets ofC such thatL = ⋂

M ∈ F . By Lemma 8.1 we have↑p(L) =⋂↑p(M) | M ∈ M. SinceK is Scott-open, there existsM ∈ M such that↑p(M) ∈ K,i.e.,M ∈F . We conclude thatF is Scott-open.

By the so-called Lawson duality for locally compact sober spaces, we have that tU = ⋃

F is open inC [15, Theorem IV-2.18]. From the definition ofU andV , we haveU ⊆ p−1(V ). Conversely ifx ∈ p−1(V ), thenp(x) ∈ V = ⋃

K. Thusp(x) ∈ K for someK ∈ K. SinceK is saturated, we have↑p(x) ⊆ K, and sinceK is a filter we conclude tha↑p(x) ∈ K. Observing that↑p(x) = ↑p(↑x), we conclude that↑x ∈ F . It follows thatx ∈ U . Thusp−1(V ) = U is open inC. It follows thatV is open in the core compactgenerated topology.

We consider the converse direction.

Lemma 8.4. Let X be sober and letU be a saturated subset. Then the compact satursubsets ofB form a Scott-open filter with unionU if and only if for any filtered familyK of compact saturated sets with intersection contained inU , then some member ofK iscontained inU .

Proof. That the first condition implies the second is immediate from the definition ofilter being Scott-open. Conversely it is also immediate that the compact saturated scontained inU form a filter (under reverse inclusion). Since point saturates belong tofamily for all points inU , the union isU . The Scott-openness of the filter follows direcfrom the last condition. Lemma 8.5. LetX be a sober space.

(i) The compact saturated sets contained in a given open setU form a Scott-open filter inthe semilattice of compact saturated sets(ordered by reverse inclusion).

(ii) Suppose that each compact saturated subset ofX is contained in a saturated set fowhich the relative topology is core compactly generated. Then the compact satsets contained in a core compactly generated open set form a Scott-open ficompact saturated sets.

Proof. (i) This follows immediately from the equivalence of the preceding lemma andHofmann–Mislove machinery for sober spaces [15, Theorem II-1.21].

(ii) Let U be open in the core compactly generated topology. LetK be a descendinfamily of compact saturated sets with intersectionE ⊆ U . By the preceding lemma, w


need only check thatK ⊆ U for someK ∈ K. Pick K0 ∈ K andA ⊇ K0 such that the

ceopen

r the

subsetactlynd

rjection

e

s

d

loods

of

relative topology onA is core compactly generated. The relative topology onA is sobersinceA is saturated [15, Exercise O-5.16]. We apply part (i) of this lemma to the spaA,the open setU ∩ A, which is a core compactly generated open subset and hence ansubset ofA, and the collectionK ∈ K: K ⊆ K0 and conclude thatK ⊆ U ∩ A ⊆ U forsomeK ∈K. Thus the filter in question is Scott-open.

Lemmas 8.3 and 8.5 yield the following result, which in turn provides an answer foquestion beginning this section.

Theorem 8.6. Let X be a sober space, and suppose that each compact saturatedof X is contained in a saturated set for which the relative topology is core compgenerated. Then a subset ofX is open in the core compactly generated topology if aonly if it can be written as a union of a Scott-open filter of compact saturated sets.

It follows from the preceding results that for a core compactly generated sober spaceX,the map that assigns to a Scott-open filter of compact saturated sets its union is a sufrom the set of all such Scott-open filters onto the lattice of open sets ofX. What is notclear is whether this map is injective. We consider sufficient conditions for this to be thcase. In this case one obtains a dual Hofmann–Mislove Theorem.

Theorem 8.7. LetX be a sober space that is core compactly generated(and hence satisfiethe hypotheses of the preceding theorem). Suppose that for each compact subsetK of X

and each open setU containingK, there exists a locally compact spaceC, a continuousfunctionp :C → X, and a compact subsetA of C such thatK ⊆ ↑p(A) ⊆ U . Then everyScott-open filter in the semilattice of compact saturated sets(ordered by reverse inclusion)is the filter of all compact saturated subsets of its union, which is an open set.

Proof. Let K be a Scott-open filter of compact saturated subsets ofX. In view of theremarks before the proposition and the earlier results, it remains to show thatK consists ofall compact saturated subsets of the open setU = ⋃

K. Thus letL be a compact saturatesubset that is contained inU . By hypothesis there exists a locally compact spaceC,a continuous mapp :C → X, and a compact subsetA of C such thatL ⊆ ↑p(A) ⊆ U .

For eachy ∈ p−1(U), we havep(y) ∈ U and hencep(y) ∈ K for someK ∈ K.Thus↑p(y) ⊆ K, and hence↑p(y) ∈ K, since the latter is a filter. It follows from locacompactness that↑y is the filtered intersection of all compact saturated neighbourhof y. By Lemma 8.1 we conclude that

↑p(y) = ↑p(↑y) =⋂↑p(B) | B is a compact saturated neighbourhood ofy

.

SinceK is Scott-open, it follows that↑p(B) ∈ K for some compact neighbourhoodBof y. Thus for everyy ∈ A, we can choose a compact neighbourhoodBy of y such that↑p(By) ∈K. Finitely many of these coverA, and hence the finite union of the saturatestheir images containsL. It follows thatL ∈K.


Corollary 8.8. LetX be a compactly generated Hausdorff space, or more generally a corellyted setson,

locallyt set offrom the

sets.of a

logy,for the

ated.

set.

rthe

then

-g of

n–

compactly generated sober space in which every compact subset is contained in a locacompact subspace. Then every Scott-open filter in the semilattice of compact satura(ordered by reverse inclusion) is the filter of all compact saturated subsets of its uniwhich is an open set.

Proof. A Hausdorff space is always sober, each compact subset is Hausdorff, hencecompact, and compactly generated implies core compactly generated. Thus the firshypotheses are a special case of the second. The second case follows immediatelypreceding theorem.Definition 8.9 (Co-sober space). A compact saturated subspaceA of a topological spaceXis k-irreducibleif it cannot be written as the union of two proper compact saturated subWe call a spaceco-soberif ever k-irreducible compact saturated set is of the saturatepoint.

(This property is loosely connected to the notion of sobriety for the cocompact topobut we do not pursue this topic here.) The next theorem gives alternative conditionsdual Hofmann–Mislove Theorem to hold.

Theorem 8.10. Let X be a sober and co-sober space that is core compactly generThen every Scott-open filter in the semilattice of compact saturated sets(ordered by reverseinclusion) is the filter of all compact saturated subsets of its union, which is an open

Proof. As in the proof of Theorem 8.7 we need only show that ifK be a Scott-open filteof compact saturated subsets ofX, thenK consists of all compact saturated subsets ofopen setU = ⋃

K.Suppose that there exists some compact saturated setK ⊆ U that is not inK. We

consider a maximal chain of compact saturated subsets such thatK is a member of thechain and each member of the chain is contained inU but does not belong toK. By themachinery of the Hofmann–Mislove Theorem, it follows that the intersectionA is a non-empty compact saturated set, andA does not belong toK since the latter is Scott-open.

If we could writeA as the union of two strictly smaller compact saturated sets,each of these would belong toK (by maximality of the chain), and henceA would belongto K (by the filter property). We conclude thatA is k-irreducible. By co-sobrietyA = ↑x

for somex ∈ U . But thenx is in some member ofK, and thus↑x = A ∈ K since the latteris a filter. This contradiction establishes the proof.

We recall the basics of Lawson duality (see [15, Section IV-2]). LetP be a directedcomplete partially ordered set. We define theLawson dualP ′ to be the collection of Scottopen filters ofP ordered by inclusion, which is again a dcpo. If the natural embeddinP into P ′′ is an order isomorphism, then we say thatP andP ′ are Lawson duals.

Suppose thatX is a sober space andO(X) is the lattice of open sets. By the HofmanMislove Theorem the Scott-open filters onO(X) may be identified withK(X), the


compact saturated subsets ofX ordered by reverse inclusion, by associating to each Scott-

setsn, morect

ate

setse

enera-ction

t

an beionally

subsetscott-

lity

open filter its intersection. With respect to this identification, the mapping fromO(X) tothe double dual sends an open setU to the Scott-open filter of all compact saturated subof U . Thus duality holds if and only if every Scott-open filter inK(X) has union an opesubset ofX and consists of all compact saturated subsets contained in this union, orbriefly said,O(X) is the Lawson dual ofK(X). This is always the case for locally compasober spaces, as worked out in [15, Section IV-2]. This was further extended in [16], but theresults there are rather technical. The preceding results of this section give as an immedicorollary a further generalization that is much less technical to derive.

Corollary 8.11. LetX be a sober core compactly generated space. ThenO(X) andK(X)

are Lawson duals(via the standard constructions of associating compact saturatedand open sets with Scott-open filters) if either of the following additional conditions arsatisfied:

(i) X is co-sober;(ii) for each compact subsetK of X and each open setU containingK, there exists a

locally compact spaceC, a continuous functionp :C → X, and a compact subsetA

of C such thatK ⊆ ↑p(A) ⊆ U .

We consider the converse problem: Does Lawson duality imply core-compact gtion? We derive a condition that additionally provides an alternative internal construof the core compactly generated topology.

Theorem 8.12. Let X be a sober space with the following property: for every compacsaturated subsetK of X there exists a probep :C → X, whereC is locally compact andsober, such that for any compact saturated subsetL of K, we havep−1(L) is compact andL = ↑p(p−1(L)). Then the core compact open sets are precisely those sets that cobtained as the union of a Scott-open filter of compact saturated sets. Hence, if additK(X) andO(X) are Lawson duals, thenX itself is core compactly generated.

Proof. Lemma 8.3 yields the implication in one direction. Conversely letW be a set that isopen in the core compactly generated topology. The collection of compact saturatedcontained inW is easily seen to be a filter (under reverse inclusion). To show it is Sopen, we consider a filtered collectionF of compact saturated sets such that

⋂F ⊆ W .

Pick K ∈ F and pick a probep as in the hypothesis. We may without loss of generarestrict the filtered collectionF to those contained inK. Then

p−1(L) | L ∈ F

is a filtered collection of compact saturated sets inC. SinceC is sober,p−1(W) is open inC, and

⋂p−1(L) | L ∈ F = p−1(⋂

F) ⊆ p−1(W), we conclude thatp−1(L) ⊆ p−1(W)

for someL ∈F . It follows that

L ⊆ ↑p(p−1(L)

) ⊆ p(p−1(W)

) ⊆ W.


If K(X) andO(X) are Lawson duals, then the union of any Scott-open filter of compactall core

viousein a

pact

erated

nerated

leof

tly7.2eserve

ces

ober?ctly

saturated sets must be an open set. But we have just seen that such unions yieldcompactly generated open sets. Hence the two topologies agree.Corollary 8.13. A Hausdorff spaceX is compactly generated if and only ifK(X) andO(X) are in Lawson duality.

Proof. A Hausdorff space is sober and easily seen to satisfy the condition of the pretheorem. Hence it is core compactly generated ifK(X) andO(X) are Lawson duals. Thconverse follows from Corollary 8.11 since condition (ii) of that corollary is satisfiedHausdorff space.

9. Open problems

We have previously mentioned Isbell’s example [20] of a space that is locally combut not generated by the locally compact Hausdorff spaces.

Problem 9.1. Is there an example of a core compactly generated space that is not genby the locally compact spaces?

It is known a Hausdorff space has the same compact sets as its compactly gecoreflection.

Problem 9.2. When doX andCX have the same compact sets?

We have seen that for any twoC-generated (and henceE-generated) spacesX andY , the productsX ×C Y andX ×E Y coincide, whereE is the class of all exponentiabspaces. Thus, there is a single construction that produces the products of all categoriesC-generated spaces.

Problem 9.3. Is there an intrinsic characterization of theC-product?

In Example 5.7, it is shown that the inclusion of sequential spaces in core compacgenerated spaces does not preserve uncountable products. On the other hand, Theoremshows that the inclusion of QCB in core compactly generated spaces does prcountable limits.

Problem 9.4. Does the inclusion of sequential spaces in core compactly generated spapreserve countable products? Does it preserve equalizers?

The last three problems relate to the concerns of Section 8.

Problem 9.5. Is the core compactly generated topology of a sober space again sDually, is the sobrification of a core compactly generated space again core compagenerated?


Problem 9.6. In a sober spaceX, is every core compactly generated open set the union of

ar on

ity

fd-

id

dge

ott

al

1–

and

ted

100

6)

ces,

in,

n,

ries

a Scott-open filter of compact saturated sets?

Problem 9.7. Is a sober space co-sober?

References

[1] M. Barr, Building closed categories, Cahiers Topologie Géom. Différentielle 19 (2) (1978) 115–129.[2] A. Bauer, A relationship between equilogical spaces and type two effectivity, Dagstuhl Semin

Computability and Complexity in Analysis, 2001, Math. Logic Quart. 48 (suppl. 1) (2002) 1–15.[3] A. Bauer, L. Birkedal, D.S. Scott, Equilogical spaces, Theoret. Comput. Sci., in press.[4] F. Borceux, Categories and structures, in: Handbook ofCategorical Algebra, vol. 2, Cambridge Univers

Press, Cambridge, 1994.[5] R. Brown, Some problems of algebraic topology: A study of function spaces, function complexes and

complexes, Ph.D. Thesis, New College, Oxford University, December 1961.[6] R. Brown, Function spaces and product topologies, Quart. J. Math. Oxford Ser. (2) 15 (1964) 238–250.[7] R. Brown, A Geometric Accountof General Topology, Homotopy Types and the Fundamental Groupo

Topology, second ed., Ellis Horwood, Chichester, 1988.[8] B.J. Day, A reflection theorem for closedcategories, J. Pure Appl. Algebra 2 (1) (1972) 1–11.[9] B.J. Day, G.M. Kelly, On topological quotient maps preserved by pullbacks or products, Proc. Cambri

Philos. Soc. 67 (1970) 553–558.[10] S. Dolecki, G.H. Greco, A. Lechicki, When do theupper Kuratowski topology (homeomorphically, Sc

topology) and the co-compact topology coincide?, Trans. Amer. Math. Soc. 347 (1995) 2869–2884.[11] J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.[12] M.H. Escardó, Topology of data types and computability concepts, Electronic Notes in Theoretic

Computer Science, in press.[13] M.H. Escardó, R. Heckmann, Topologies on spacesof continuous functions, Topology Proc. 26 (2) (200

2002) 545–564.[14] D. Gale, Compact sets of functions and function rings, Proc. Amer. Math. Soc. 1 (1950) 303–308.[15] G. Gierz, K.H. Hofmann, K. Keimel, J.D. Lawson, M. Mislove, D.S. Scott, Continuous Lattices

Domains, Cambridge University Press, Cambridge, 2003.[16] K.H. Hofmann, J.D. Lawson, On the order-theoretical foundation of a theory of quasicompactly genera

spaces without separation axiom, J. Austral. Math. Soc. Ser. A 36 (2) (1984) 194–212, MR 85g:06002.[17] J.M.E. Hyland, Filter spaces and continuous functionals, Ann. of Math. Logic 16 (1977) 101–143.[18] J. Isbell, Function spaces and adjoints, Math. Scand. 36 (1975) 317–339.[19] J. Isbell, General function spaces, products and continuous lattices, Math. Proc. Cambridge Philos. Soc.

(1986) 193–205.[20] J. Isbell, A distinguishing example ink-spaces, Proc. Amer. Math. Soc. 100 (1987) 593–594.[21] J.L. Kelley, General Topology, Van Nostrand, New York, 1955.[22] S. Mac Lane, Categories for the Working Mathematician, Springer, Berlin, 1971.[23] G. Markowsky, Chain-complete posets and directed sets with applications, Algebra Universalis 6 (197

53–68.[24] M. Menni, A. Simpson, Topological and limit-space subcategories of countably-based equilogical spa

Math. Structures Comput. Sci. 12 (6) (2002) 739–770.[25] E. Michael,ℵ0-spaces, J. Math. Mech. 15 (1966) 983–1002.[26] D. Normann, Recursion on the Countable Functionals, Lecture Notes in Math., vol. 811, Springer, Berl

1980.[27] M. Schröder, Admissible representations for continuous computations, Ph.D. Thesis, Universität Hage

2002.[28] M. Schröder, Extended admissibility, Theoret. Comput. Sci. 284 (2) (2002) 519–538.[29] F. Schwarz, Powers and exponential objects in initiallystructured categories and applications to catego

of limit spaces, Quaestiones Math. 6 (1983) 227–254.


[30] F. Schwarz, S. Weck, Scott topology, Isbell topology and continuous convergence, in: R.-E. Hoffmann,es

ALGI

2–198

71)

K.H. Hofmann (Eds.), Continuous Lattices and Their Applications, Bremen, 1982, New York, Lecture Notin Pure Appl. Math., vol. 101, Marcel Dekker, New York, 1985, pp. 251–273.

[31] A. Simpson, Towards a convenient category of topological domains, in: Proceedings of 13thWorkshop, RIMS, Kyoto University, 2003.

[32] E. Spanier, Infinite symmetric products, function spaces, and duality, Ann. of Math. (2) 69 (1959) 14[Erratum, 733].

[33] N.E. Steenrod, A convenient category of topological spaces, Michigan Math. J. 14 (1967) 133–152.[34] R.M. Vogt, Convenient categories of topological spaces for homotopy theory, Arch. Math. (Basel) 22 (19

545–555.[35] S. Weingram, Bundles, realizations andk-spaces, Ph.D. Thesis, Princeton University, 1961.

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