Hereditary Coreflective Subcategories of the Categories of Tychonoff and Zero-Dimensional Spaces

Appl Categor Struct (2013) 21:671–679DOI 10.1007/s10485-012-9285-4

Hereditary Coreflective Subcategories of the Categoriesof Tychonoff and Zero-Dimensional Spaces

Juraj Cincura

Received: 8 December 2011 / Accepted: 24 April 2012 / Published online: 16 May 2012© Springer Science+Business Media B.V. 2012

Abstract In this paper the structure of hereditary coreflective subcategories in thecategories Tych of Tychonoff and ZD of zero-dimensional spaces is studied. It isshown that there are (many) hereditary additive and divisible subcategories in Tychand ZD which are not coreflective. Moreover, if A is an epireflective subcategory ofthe category Top of topological spaces which is not bireflective and B is an additiveand divisible subcategory of A which is not coreflective, then the coreflective hull ofB in A is not hereditary. It is also shown, in the case of Tych under Martin’s axiom orunder the continuum hypothesis, that if B is a hereditary coreflective subcategory ofTych (ZD), then either the topologies of all spaces belonging to B are closed undercountable intersections or it contains all Tychonoff spaces (zero-dimensional spaces)with Ulam nonmeasurable cardinality.

Keywords Coreflective subcategory · Epireflective subcategory · Hereditarycoreflective subcategory · Hereditary additive and divisible subcategory

Mathematics Subject Classifications (2010) 18D15 · 54B30

1 Introduction

In the paper [8] Herrlich and Hušek suggested to investigate the structure of heredi-tary coreflective subcategories of the category Top of topological spaces. The authorand Sleziak published in [3, 4, 13] several results concerning hereditary coreflective

The author acknowledges the support of the VEGA Grant 1/0588/09.

J. Cincura (B)KAGDM FMFI UK, Mlynská dolina,842 48 Bratislava, Slovakiae-mail: [email protected]

672 J. Cincura

subcategories not only in Top but also in epireflective and quotient-reflective subcat-egories of Top. Sleziak (see [14]) extended these results to hereditary additive anddivisible (H AD) subcategories of epireflective subcategories of Top contained inHaus. Note, that in the category Top and also in any quotient-reflective subcategoryof Top coreflective subcategories are precisely additive and divisible subcategories.In epireflective subcategories of Top, any coreflective subcategory is additive anddivisible but the converse is not true (e.g. the subcategory of k-spaces is additive anddivisible without being coreflective in Tych). H AD-subcategories in epireflectivesubcategories of Top are in some sense closer to hereditary coreflective subcategoriesof Top than hereditary coreflective subcategories. As it is shown in [14], if A is anepireflective subcategory of Top with A ⊆ Haus, then the assignment B �→ B ∩ Ais a bijection from the collection of all hereditary coreflective subcategories of Topeach of which contains all finitely generated spaces to the collection of all H AD-subcategories of A. The notion of H AD-subcategory is also more general thanthe notion of hereditary coreflective subcategory. On the other hand, coreflectivesubcategories of any epireflective subcategory of Top are both complete and co-complete, while AD-subcategories may be neither complete nor cocomplete (e.g.the subcategory of k-spaces in Tych) so that coreflective subcategories are in thissense more convenient than AD-subcategories and it is useful to know whether an(H)AD-subcategory of Tych (ZD) is coreflective or not.

The aim of this paper is to investigate hereditary coreflective subcategories in thecategories Tych of Tychonoff and ZD of zero-dimensional spaces and their relationto H AD-subcategories in these categories. First we show that there exist non-trivialhereditary coreflective subcategories in Tych and ZD. Then we prove that if ahereditary coreflective subcategory B of Tych (ZD) contains all countable Tychonoff(zero-dimensional) spaces, then it contains all Tychonoff (zero-dimensional) spaceswith nonmeasurable cardinality (in the sense of Ulam). As a consequence we obtainmany examples of H AD-subcategories of Tych (ZD) that are not coreflective. Wealso show, in the case of Tych under MA (Martin’s axiom) or under CH (thecontinuum hypothesis), that if B is a hereditary coreflective subcategory of Tych(ZD), then either the topologies of all spaces belonging to B are closed undercountable intersections or B contains all Tychonoff (zero-dimensional) spaces withnonmeasurable cardinality. Finally we prove that if A is an epireflective subcategorywhich is not bireflective and B is an AD-subcategory of A which is not coreflective,then the coreflective hull of B in A is not hereditary.

2 Preliminaries and Notation

Throughout this paper all subcategories are supposed to be full and isomorphism-closed and every coreflective subcategory is supposed to contain a non-empty space(and, consequently, to be bicoreflective).

We denote the category of topological spaces (and continuous maps) by Top, thecategory of Hausdorff spaces by Haus, the category of Tychonoff spaces by Tych, thecategory of zero-dimensional spaces by ZD and the discrete space on the set {0, 1} byD(2).

A subcategory A of Top is epireflective in Top if and only if it is closed underthe formation of topological products and subspaces. If A is a quotient-reflective

Hereditary Coreflective Subcategories of Tych and ZD 673

subcategory of Top (i. e. all A-reflections are (usual) quotient maps), then an A-morphism f is an extremal epimorphism in A if and only if f is a (usual) quotientmap in Top.

Let A be an epireflective subcategory od Top. It is well known (see e.g. [9, 10])that a subcategory B of A is coreflective in A if and only if it is closed under theformation of A-coproducts (i. e. topological sums) and extremal A-quotients. If S isa subcategory of A or a class of A-objects, then the coreflective hull CA(S) of S inA consists precisely of all extremal A-quotients of coproducts of members of S . Acoreflective subcategory B of A is said to be generated by a class S provided that B =CA(S).

A subcategory B of Top is said to be additive if it is closed under the formationof topological sums and hereditary if it is closed under the formation of subspaces. Asubcategory B of A, A being a subcategory of Top, is said to be divisible in A if forany (usual) quotient map q : X → Y with X ∈ B and Y ∈ A, the space Y belongs toB. A subcategory B of A which is additive and divisible in A is called briefly AD-subcategory in A. If B is moreover hereditary we say that it is an H AD-subcategoryin A. If A = Top or A is a quotient-reflective subcategory of Top, then the notionof AD-subcategory (H AD-subcategory) coincides with the notion of (hereditary)coreflective subcategory in A.

Recall that a topological space X is said to be a prime space provided that Xhas exactly one non-isolated point. Every prime T1-space is zero-dimensional. If Xis a topological space and a ∈ X, then by Xa we denote the space constructed bymaking each point of X, other than a, isolated, retaining original neighbourhoods ofa. The space Xa is called the prime factor of X at a. Obviously, any prime factor of atopological space is either a prime space or a discrete space and for any topologicalspace X there exists a quotient map from the topological sum of its prime factorsto X.

Cardinals are initial ordinals, by nonmeasurable cardinal we mean Ulam nonmea-surable cardinal.

Undefined terminology can be found in [1, 5].

3 Results

First we show that there are non-trivial hereditary coreflective subcategories in thecategory Tych and in the category ZD.

Let α be a regular cardinal. Denote by Topα the subcategory of Top consistingof all spaces X such that for any nonempty family U of open subsets of X withcard(U) < α the set

⋂{U : U ∈ U} is open in X (clearly, Topω0= Top). It is well

known (see e.g. [7]) that any Topα is a hereditary coreflective subcategory of Top.Put Tychα = Topα ∩ Tych, ZDα = Topα ∩ ZD.

Proposition 1 If α is a regular cardinal, then ZDα is a hereditary coref lective subcate-gory in ZD.

Proof It is easy to verify that the Topα-coreflection of of every zerodimensionalspace is a zerodimensional space. ��

674 J. Cincura

Proposition 2 For any regular cardinal α, the category Tychα is a hereditarycoref lective subcategory in Tych. If α ≥ ω1, then Tychα = ZDα .

Proof Let α ≥ ω1 and X ∈ Tych. Then there exists a topological embedding j :X → IA where I = [0, 1] and A is a suitable set. Let the functor C : Top → Topα

be the coreflector corresponding to Topα . Then C( j) : C(X) → C(IA) is a regularmonomorphism in Topα . Since Topα is hereditary and contains indiscrete spacesregular monomorphisms in Topα are precisely topological embeddings. The spaceC(I) is discrete and the space C(I)A is zero-dimensional. Because C(IA) = C(C(I)A)

(C preserves products) we obtain that C(IA) is zero-dimensional (it follows fromProposition 1). Consequently, C(X) is zero-dimensional and Tychα = ZDα . ��

Remark 1 The category B = Tychω1= ZDω1 is coreflective both in Tych and ZD. It

is easy to see that if C is a subcategory of B, then C is coreflective in Tych if and onlyif it is coreflective in ZD.

Any epireflective subcategory A of Top is well-powered and complete andtherefore, by [10, Theorem 34.1], it is an (extremal epi, mono)-category. This yieldsthe following characterization of extremal epimorphisms in A (well known in thecase of quotient maps in Top).

Proposition 3 Let A be an epiref lective subcategory of Top and f : X → Y be an A-morphism. Then f : X → Y is an extremalA-epimorphism if and only if it is surjectiveand for any map g : Y → Z with Z ∈ A the map g is continuous whenever the mapg ◦ f is continuous.

Proof Suppose that f : X → Y is an extremal A-epimorphism. Since the subspacef (X) of Y belongs to A we obtain f (X) = Y. Let g : Y → Z , Z being an A-space,be a map such that g ◦ f is continuous. Then there exists an extremal A-epimorphisme : X → B and an A-monomorphism m : B → Z for which g ◦ f = m ◦ e. By the(extremal epi, mono)-diagonalization property there exists an A-morphism h : Y →B such that m ◦ h = g. Hence, the map g is continuous.

Conversely, let f : X → Y fulfil the given conditions, and f = m ◦ g where g is anA-morphism and m is an A-monomorphism. Then m is bijective and m−1 ◦ f = g iscontinuous. Hence, m−1 is continuous and m is an isomorphism. ��

Recall that D(2) is the discrete space on the set {0, 1}.

Corollary 1 A continuous map f : X → Y in Tych (ZD) is an extremal epimorphismin Tych (ZD) if and only if it is surjective and for every map g : Y → R (g : Y →D(2)) the map g is continuous whenever the map g ◦ f is continuous.

Next we prove that if a hereditary coreflective subcategory B of Tych (ZD)

contains all countable Tychonoff (zero-dimensional) spaces, then B contains allTychonoff (zero-dimensional) spaces with nonmeasurable cardinality.


Lemma 1 Let B be a hereditary coref lective subcategory of Tych (ZD), X be aTychonof f (zero-dimensional) space with card(X) = α and γ = 2α . If the space D(2)γ

belongs to B, then X ∈ B.

Proof Every prime factor of X can be embedded into D(2)γ and therefore belongsto B. Hence, X ∈ B. ��

A topological space X is said to be a cR-space (see e.g. [16]) (a cD(2)-space) pro-vided that any map f : X → R ( f : X → D(2)), whose restriction to each countablesubspace of X is continuous, is continuous. Obviously, if a space X is a cR-space,then it is a cD(2)-space. Denote by CR (CD(2)) the subcategory of Tych (ZD) consistingof all cR-spaces (cD(2)-spaces). Clearly, CR (CD(2)) contains all countable Tychonoff(zero-dimensional) spaces.

Proposition 4

(1) Let X ∈ Tych (X ∈ ZD), S be the family of all countable subspaces of X andp : ∑

S∈S S → X be the natural projection. Then X is a cR-space (cD(2)-space) ifand only if p is an extremal epimorphism in Tych (ZD).

(2) The category CR (CD(2)) is the coref lective hull of the class of all countableTychonof f (zero-dimensional) spaces in the category Tych (ZD).

Proof (1) follows from Corollary 1, (2) is obvious. ��

In the paper [15] Uspenskii proved the following result:

Proposition 5 [15] Let (Xa : a ∈ A) be a family of f irst countable spaces and X =∏a∈A Xa. If card(A) is nonmeasurable, then X is a cR-space.

As a consequence we obtain that if γ is a nonmeasurable cardinal, then D(2)γ is acR-space.

Theorem 1 If a hereditary coref lective subcategory B of Tych (ZD) contains allcountable Tychonof f (zero-dimensional) spaces, then B contains all Tychonof f (zero-dimensional) spaces with nonmeasurable cardinality.

Proof Obviously, CR ⊆ B (CD(2) ⊆ B). If α is a nonmeasurable cardinal, then thecardinal γ = 2α is also nonmeasurable and therefore D(2)γ , being a cR-space,belongs to B. The rest follows from Lemma 1. ��

Let α be an infinite cardinal. Denote by Genα the coreflective hull of the classof all spaces X with card(X) ≤ α in Top (a space X belongs to Genα if and only ifthe tightness of X is at most α). It is well known (see e.g. [7]) that for any infinitecardinal α the category Genα is a hereditary coreflective subcategory of Top. LetTGα = Tych ∩ Genα and ZGα = ZD ∩ Genα . Since every Genα (being hereditaryand coreflective in Top) is an H AD-subcategory of Top we obtain that every TGα isan H AD-subcategory of Tych and every ZGα is an H AD-subcategory of ZD (TGα

(ZGα) is the AD-hull of the class of all Tychonoff (zero-dimensional) spaces X withcard(X) ≤ α in Tych (ZD)).

676 J. Cincura

Clearly, each TGα (ZGα) contains all countable Tychonoff (zero-dimensional)spaces. Hence, as a consequence of Theorem 1, we obtain:

Proposition 6 If α is an inf inite nonmeasurable cardinal, then TGα (ZGα) is notcoref lective in Tych (ZD) and the coref lective hull of TGα in Tych (the coref lectivehull of ZGα in ZD) is not hereditary.

Example 1 We show that if α is an infinite nonmeasurable cardinal, then the categoryTGα is neither complete nor cocomplete.

Let α be an infinite nonmeasurable cardinal. Then the successor α+ of α is aregular cardinal and the space C(α+), defined on the set α+ ∪ {α+} such that a set U isopen in C(α+) if and only if U ⊆ α+ or card(α+ \ U) < α+, does not belong to TGα .Let β = 2(α+). Then, clearly, D(2)β does not belong to TGα (C(α+) is embeddableinto D(2)β) and it is a cR-space (β is nonmeasurable). We want to show that the βthTGα-power of the space D(2) does not exist. Suppose, on the contrary, that thereexists the TGα-power ((D(2)β)α, (qλ)λ∈β) of the space D(2). For each λ ∈ β denote bypλ : D(2)β → D(2) the corresponding natural projection. Then there exists exactlyone continuous map q : (D(2)β)α → D(2)β such that pλ ◦ q = qλ for all λ ∈ β. LetA be a countable subspace of the space D(2)β . Then for every λ ∈ β the mappλ |A: A → D(2) is continuous, the space A belongs to TGα and therefore thereexist precisely one continuous map ϕA : A → (D(2)β)α such that qλ ◦ ϕA = pλ |A forall λ ∈ β. Define the map ϕ : D(2)β → (D(2)β)α by ϕ(x) = ϕ{x}(x). It is easy to seethat q ◦ ϕ = idD(2)β . Obviously, if A is a countable subspace of D(2)β and x ∈ A,then ϕA(x) = ϕ{x}(x) = ϕ(x). Hence, the map ϕ : D(2)β → (D(2)β)α is continuous onevery countable subspace of the space D(2)β . Since the space D(2)β is a cR-spaceand the space (D(2)β)α is a Tychonoff space we obtain that the map ϕ is continuous.Consequently, the space D(2)β is homeomorphic to the subspace C = ϕ(D(2)β) ofthe space (D(2)β)α . Since C belongs to TGα this yields a contradiction. Thus, thecategory TGα fails to have products.

It is easy to see that the category TGα is co-(well-powered) (TGα-epimorphismsare dense continuous maps) and strongly well-powered (i. e., for every set-indexedfamily (Ya)a∈J of TGα-spaces there exists a set {Xc : c ∈ I} of TGα-spaces such thatif (X, (ma : X → Ya)a∈J) is a monosource in TGα , then there is c ∈ I for which X isisomorphic with Xc). Therefore, according to [10, Theorem 23.13], since the categoryTGα is not complete it is not cocomplete.

Similarly, it can be shown that if α is an infinite nonmeasurable cardinal, then thecategory ZDα is neither complete nor cocomplete.

The assertion concerning the coreflective hull of TGα in previous propositioncan be generalized to an analogous result for AD-subcategories of any epireflectivesubcategory of Top which is not bireflective.

Lemma 2 If A is an epiref lective subcategory of Top, B is an AD-subcategory of Aand P is a prime space belonging to the coref lective hull CA(B) of B in A, then P ∈ B.

Proof Observe that if an epireflective subcategory A of Top contains a prime spaceP, then it contains the space D(2), all zero-dimensional spaces and, consequently,all prime T1-spaces. If P is not a T1-space then A contains the Sierpinsky doubleton


S(2) (there exists a section S(2) → P) and this yields that A is the category of allT0-spaces or A = Top and B is coreflective in A.

Let P ∈ CA(B). If P is not a T1-space, then CA(B) = B and P ∈ B. If P is a primeT1-space, then (because B is additive) there exists an extremal A-epimorphism q :X → P with X ∈ B. Let q = m ◦ p where p : X → Y is a (usual) quotient map andm : Y → P is an injective continuous map. Since P is a prime T1-space and m isinjective and continuous, the space Y is a prime T1-space or a discrete space. Hence,Y ∈ A and, consequently, Y ∈ B (B is divisible in A). Moreover, p is an A-morphism,m is an A-monomorphism and therefore (because q is an extremal A-epimorphism)m is an isomorphism. Thus, P ∈ B. ��

Theorem 2 Let A be an epiref lective subcategory of Top which is not biref lectiveand B be an AD-subcategory of A which is not coref lective. Then the coref lective hullCA(B) of B in A is not hereditary.

Proof Suppose that CA(B) is hereditary. If X ∈ CA(B), then, according to [4,Theorem 1], all prime factors of X belong to CA(B) and by Lemma 2 they belongto B. As X is a quotient space of the topological sum of its prime factors we obtainthat X ∈ B. Hence CA(B) = B and we have a contradiction. ��

Finally we show that if B is a hereditary coreflective subcategory of ZD, then Bcontains all zero-dimensional spaces with nonmeasurable cardinality or B ⊆ ZDω1 .In the case of the category Tych we prove an analogous result provided that thecardinal c = 2ω0 is not a sequential cardinal. This assumption is fulfilled, for instance,under MA or under CH.

Denote by C(ω0) the Alexandroff compactification of the discrete space ω0

defined on the set ω0 ∪ {ω0} (a set U is open in C(ω0) if and only if U ⊆ ω0 orω0 \ U is finite). The following lemma easily follows from [14, Proposition 6.11 andLemma 6.2].

Lemma 3

(1) If P is a prime T2-space which does not belong to Tychω1(= ZDω1), then there

exists a quotient map q : P → P1 where P1 is a prime T2-space def ined on theset ω ∪ {ω0} such that ω0 is not isolated in P1.

(2) If P1 is a prime T2-space on the set ω ∪ {ω0} such that the point ω0 is not isolatedin P1, then C(ω0) belongs to the AD-hull of the space P1 in Tych (ZD).

Denote by TSeq (ZSeq) the category of all Tychonoff sequential (zero-dimensional sequential) spaces. It is well known that TSeq (ZSeq) is the AD-hullof the space C(ω0) in Tych (ZD).

Lemma 4 If B is a hereditary coref lective subcategory of Tych (ZD) such that B ⊆Tychω1

(B ⊆ ZDω1), then TSeq ⊆ B (ZSeq ⊆ B).

Proof According to [4, Theorem 1] there exists a class S of prime spaces such that Bis the coreflective hull of S in Tych (ZD). Since Tychω1

(ZDω1) is coreflective, thereexists P ∈ S with P /∈ Tychω1

(P /∈ ZDω1). By Lemma 3 we obtain that C(ω0) belongsto B and therefore TSeq ⊆ B (ZSeq ⊆ B). ��

678 J. Cincura

A topological space X is said to be an sR-space (an sD(2)-space) provided that everysequentially continuous map f : X → R ( f : X → D(2)) is continuous. Evidently,every sR-space is an sD(2)-space. Denote by TSeq

R(ZSeqD(2)) the category of all

Tychonoff sR-spaces (zero-dimensional sD(2)-spaces). Using Corollary 1 it can beeasily proved that the following proposition is valid.

Proposition 7

(1) The category TSeqR

(ZSeqD(2)) is a coref lective subcategory of Tych (ZD).(2) TSeq

R(ZSeqD(2)) is the coref lective hull of TSeq in Tych (ZSeq in ZD).

From [2, Corollary 2.5] or [12, Theorem 3.2 (iii)] it follows that if a cardinal γ isnonmeasurable, then the space D(2)γ is an sD(2)-space.

Theorem 3 Let B be a hereditary coref lective subcategory of ZD. If B ⊆ ZDω1 , thenB contains all zero-dimensional spaces with nonmeasurable cardinality.

Proof According to Lemma 4 and Proposition 7, ZSeqD(2) ⊆ B and therefore if γ

is a nonmeasurable cardinal, then D(2)γ ∈ B. The rest of the proof follows fromLemma 1. ��

Example 2 Let SSeq be the subcategory of Top consisting of all subsequential spaces(see [6]). SSeq is a hereditary coreflective subcategory of Top. The category ZSSeq =SSeq ∩ ZD is an H AD-subcategory of ZD containing the space C(ω0) and thereforeZSSeq ⊆ ZDω1 . Hence, ZSSeq is not coreflective in ZD and the coreflective hull ofZSSeq in ZD is not hereditary.

Recall (see e.g. [12]) that a cardinal α is called sequential if there exists a nonzeroreal valued function σ : P(α) → R (P(α) is the powerset of the set α) whichpreserves the convergence of sequences of subsets of α and maps finite sets to zero.It is shown in [11] that each cardinal less that the first weakly inaccesssible cardinalis not sequential. In [12] it is proved the following result:

Theorem 4 [12] Let X = ∏a∈A Xa where each Xa is a Tychonof f sR-space. If each Xa

is locally pseudocompact, then X is an sR-space if and only if card(A) is not sequential.

It is not known (in ZFC) whether the cardinal c = 2ω0 is sequential or not. But, forinstance, under Martin’s axiom (see e.g. [2, Theorem 4.1]) or under CH this cardinalis not sequential. Hence, in the following theorem, the condition “the cardinal c isnot sequential” can be replaced by Martin’s axiom or by CH.

Theorem 5 Let B be a hereditary coref lective subcategory of Tych. If the cardinalc = 2ω0 is not sequential and B ⊆ Tychω1

, then B contains all Tychonof f spaces withnonmeasurable cardinality.

Proof By Theorem 4 the space D(2)c is an sR-space and by Lemma 4 and Proposition7 it belongs to B. Therefore (by Lemma 1) B contains all countable Tychonoffspaces and this yields (by Proposition 5) that B contains all Tychonoff spaces withnonmeasurable cardinality. ��


Remark 2 The question whether there is a proper hereditary coreflective subcat-egory of the category Tych (ZD) which contains all spaces with nonmeasurablecardinality (provided that there exist measurable cardinals) is still open. If everycardinal is nonmeasurable, then, of course, the category B in Theorems 1, 3 and 5coincides with the whole category Tych or ZD, respectively.

Remark 3 The problem of describing the structure of hereditary coreflective sub-categories of Tych (ZD) contained in Tychω1

(ZDω1) remains open. Accordingto Remark 1 it suffices to investigate hereditary coreflective subcategories of ZDcontained in ZDω1 . It seems that, unlike H AD-subcategories, hereditary coreflectivesubcategories in Tych and ZD are rare (it is even possible that the category ofdiscrete spaces and the categories Tychα (ZDα) are the only hereditary coreflectivesubcategories of the category Tych (ZD)).

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Hereditary Coreflective Subcategories of the Categories of Tychonoff and Zero-Dimensional Spaces