Apartness, Topology, and Uniformity: a Constructive View

Math. Log. Quart. 48 (2002) Suppl. 1, 16 – 28

Mathematical LogicQuarterly

c© 2002 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Apartness, Topology, and Uniformity: a Constructive View

Douglas Bridgesa, Peter Schusterb, and Luminita Vıtaa,c

a Department of Mathematics and Statistics, University of Canterbury, Private Bag4800, Christchurch, New Zealand1)

b Mathematisches Institut, Ludwig–Maximilians–Universitat Munchen, Theresien-straße 39, 80333 Munchen, Germany2)

c Catedra de Matematica, Universitatea Dunarea de Jos, Str. Domneasca nr. 47,Galati, Romania3)

Abstract. The theory of apartness spaces, and their relation to topological spaces (in thepoint–set case) and uniform spaces (in the set–set case), is sketched. New notions of localdecomposability and regularity are investigated, and the latter is used to produce an exampleof a classically metrisable apartness on R that cannot be induced constructively by a uniformstructure.

Mathematics Subject Classification: 54E05, 54E17, 03F60.

Keywords: Apartness spaces, constructive.

1 Introduction

At first sight, the main constructive problem with the notion of a topological space—the definition of which is perfectly acceptable within constructive mathematics—isthat it is essentially a pointwise one. Without the additional structure provided by auniformity, a topological space can only handle pointwise continuous functions, which,even for metric spaces, generally do not provide enough information for interesting,successful computations; for the latter, we normally need the strength of uniformcontinuity,4) which arises naturally in the context of a metric space.

In this paper we first sketch the constructive theory of apartness—between pointsand sets, and between sets and sets—a very recent development which seems to hold

1)e-mail: [email protected])email: [email protected])e-mail: [email protected])Strong continuity, a slight weakening of uniform continuity, suffices for certain purposes such

as approximating intermediate values [22]. Strong continuity arises naturally when one considersapartness between sets.

c© 2002 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 0942-5616/02/4811-0016 $ 17.50+.50/0

Apartness, Topology, and Uniformity 17

considerable promise as a substitute for classical topology. We then introduce notionsof local decomposability and of regularity, showing how these tidy up a number ofloose ends of the theory. In particular, we show that our constructive theory ofapartness spaces strictly includes that of uniform spaces, by providing an exampleof a by–no–means–pathological apartness on R that cannot be induced by a uniformstructure.5)

We emphasise from the outset that to all intents and purposes, Bishop–style con-structive mathematics (CM) is just mathematics with intuitionistic logic. Proofscarried out with that logic are automatically constructive, as is shown by realizabilityinterpretations like those of Kleene [16]. Results proved with intuitionistic logiccan be translated6) mutatis mutandis into formal systems for computable analysis,such as recursive function theory or Weihrauch’s Type Two Effectivity Theory [28].Moreover, all our results and proofs are valid in classical (traditional) mathematics.So we have a trade–off: a weaker logic for more interpretations, without destroyingthe validity of our work in the traditional, classical–logic based, setting.

For background in CM the reader should consult the early chapters of [1, 3, 24, 7];for aspects of intuitionistic topology, see [23, 27].

2 Point–set apartness

We work throughout with a set7) X equipped with a binary relation �= of inequality,or point–point apartness, that satisfies the properties

x �= y ⇒ ¬ (x = y) ,x �= y ⇒ y �= x

and that is nontrivial in the sense that there exist x, y in X with x �= y.A subset S of X has two natural complementary subsets:• the logical complement ¬S = {x ∈ X : ∀y ∈ S ¬ (x = y)} ,8)

• the complement ∼S = {x ∈ X : ∀y ∈ S (x �= y)} .Initially, we are interested in the additional structure imposed on X by a relation

apart(x, S) between points x ∈ X and subsets S of X. For convenience we introducethe apartness complement

−S = {x ∈ X : apart(x, S)}5)In contrast to ‘pointless’ approaches such as the formal topology practised by Sambin and

others [19], ours keeps to the point–set perspective which, in spite of the aforementioned reservations

about pointwise continuity, is more in character with constructive mathematics a la Bishop. Apromising task, however, for future work is to link our version of constructive topology with formal

topology, and eventually to provide a point–free variant of the theory of apartness spaces. As thecurrent version of apartness space theory originates in the usual point–set theory of metric spaces,

a possible guideline is the point–free theory of metric spaces developed by Curi [13] within formaltopology.

6)The talk by Peter Lietz at the CCA ’01 Symposium in Dagstuhl gave strong evidence in

support of this claim.7)In CM, every set arrives with an equivalence relation= of equality, which mappings, relations,

subsets, propositions, etc. are expected to respect.8)To avoid ambiguity of interpretation of the symbol ¬, we adopt the convention that, for

example, ¬S ⊂ T means (¬S) ⊂ T and should not be confused with ¬ (S ⊂ T ) .

18 Douglas Bridges, Peter Schuster, and Luminita Vıta

of S; and, when A is also a subset of X, we write A− S = A ∩ −S.We call X a point–set apartness space—or, for simplicity in this section,

simply an apartness space—if the following axioms (introduced in [9]) are satisfied.A1 x �= y ⇒ apart(x, {y})A2 apart(x, A)⇒ x /∈ AA3 apart (x, A ∪B) ⇔ apart(x, A) ∧ apart(x, B)A4 x ∈ −A ⊂ ∼B ⇒ apart(x, B)A5 apart (x, A)⇒ ∀y ∈ X (x �= y ∨ apart(y, A))Under classical logic, axiom A5 holds trivially when inequality is taken as the

denial of equality. More generally, axiom A5 obtains whenever X, together with =and �=, is a discrete set—that is,

x = y ∨ x �= y.Although the real numbers with the usual metric (in)equality do not form a discreteset in CM (see, for example, [4]), the inequality between the points of any metricspace is cotransitive:

x �= z ⇒ ∀y (x �= y ∨ y �= z) .Note that axiom A5 is grounded on this property: for if, on a set with inequality, wedefine apartness by

apart(x, A)⇔ x ∈∼A,then A1–A4 trivially obtain, and A5 for a singleton set A is equivalent to �= beingcotransitive.

An example in Section 5 of [9], based on the classical Smirnov topology on [0, 1] ,shows that, constructively, axiom A5 is independent of A1–A4 even when the inequal-ity is the usual metric one. A referee has provided us with a simple example of theindependence of A5 from A1–A4 when the inequality is not the denial of equality.Take X = {0, 1, 2, 3} with the usual equality and with the inequality and apartnessdefined by

x �= y ⇔ x− y is odd,apart (x, A)⇔ x /∈ A.

Then A1–A4 hold, but we have apart (0, {2}) , ¬ (0 �= 2) , and ¬ (apart (2, {2})) , soA5 does not hold.

Note that in this example the peculiar inequality is not standard—that is, clas-sically equivalent to the denial of equality—whereas any metric inequality clearly isstandard. Most of the time we are interested in standard inequalities. In this exam-ple, as soon as we modify either = or �= in order to make the inequality standard, wecan no longer disprove A5. This is clear when = and �= are understood as identityand the negation thereof, respectively; if we decide, however, that x = y and x �= yare to stand for x − y being even and odd, respectively, then we have to take intoaccount that ∅, {0, 2}, {1, 3}, and X are the only (extensional) subsets of X.

It readily follows from A2, A5 and the definitions of the various complements that

−S ⊂∼S ⊂ ¬S .


In particular, the converse of A1 can be deduced from the axioms.We say that the point x ∈ X is near the set A ⊂ X, and we write near(x, A) , if

∀S (x ∈ −S ⇒ ∃y (y ∈ A − S)) .If near (x, A) and apart (x, A) , then there exists y ∈ A− A, which is absurd; thus

near (x, A)⇒ ¬apart (x, A) .

In fact, our notion of nearness is classically equivalent to the negation of apartnessprovided that inequality is the denial of equality. For if ¬near (x, A) , then classicallythere exists S such that apart (x, S) and A− S = ∅; whence x ∈ −S ⊂∼A (becausethen X = A ∪ ¬A and ¬A =∼A) and therefore, by axiom A4, apart (x, A) . In thecorresponding classical development [12], nearness is taken as the primitive notion,apartness is defined as the negation of nearness, and inequality is the denial of equality.As it is shown in [9], the equivalence of nearness and the negation of apartness doesnot hold constructively, since it implies the law of excluded middle in the form

¬¬P ⇒ P.

An example of an apartness structure occurs when (X, τ ) is a certain kind of T1

topological space.9) If x ∈ X and A ⊂ X, we defineapart(x, A)⇔ ∃U ∈ τ (x ∈ U ⊂ ∼A)

and then introduce near (x, A) as above. The relation apart satisfies axioms A2–A4,but to make X into an apartness space we also need to postulate axiom A5. We callthis apartness structure on X the topological apartness structure correspondingto τ.

A subset S of an apartness space X is said to be nearly open if there exists afamily (Ai)i∈I such that S =

⋃i∈I −Ai. The nearly open sets form a topology—the

apartness topology—on X for which the apartness complements form a basis.P r o p o s i t i o n 2.1. ([9], Proposition 24) Every nearly open set in a topological

apartness space is open.Since the apartness topology is T1 by its very definition, to achieve Proposition

2.1 it is necessary to assume that the original topology be T1.We say that a topological apartness space X is topologically consistent if every

open subset X is nearly open. If every topological apartness space is topologicallyconsistent, then10) we can prove the law of excluded middle in the form

¬¬P ⇒ P.

However, there is a natural condition that ensures topological consistency: an apart-ness space X is said to be locally decomposable if

(1) ∀x ∈ X∀S ⊂ X (x ∈ −S ⇒ ∃T (x ∈ −T ∧ ∀y ∈ X (y ∈ −S ∨ y ∈ T ))) .Note that local decomposability always holds in the classical development. For ifx ∈ −S and we take T =∼ −S, then

x ∈ −S ⊂∼∼ −S =∼ T,9)That is, for every x, y ∈ X with x �= y there are U,V ∈ τ with x ∈ U �� y and x �∈ V � y.10)This was shown to us in a private communication by Jeremy Clarke.


so, by axiom A4, apart (x, T ) . Also, since in the classical theory inequality is takenas the denial of equality, we then have X = −S ∪ T.

P r o p o s i t i o n 2.2. Let X be a locally decomposable apartness space. Then theapartness topology τ has the property

(2) ∀x ∈ X∀U ∈ τ (x ∈ U ⇒ ∃V ∈ τ (x ∈ V ∧ ∀y ∈ X (y ∈ U ∨ y ∈∼V ))) .Conversely, a topological apartness space (X, τ ) satisfying (2) is locally decomposable.

P r o o f. Suppose that X is locally decomposable, and denote the apartness topol-ogy on X by τ. Let U ∈ τ and let x ∈ U. Without loss of generality, we may assumethat U = −S for some S ⊂ X. Now choose T as in (1), and set V = −T ; then x ∈ V.Moreover, for each y ∈ X either y ∈ −S = U or else

y ∈ T ⊂∼∼ T ⊂∼ −T =∼ V(because −T ⊂∼ T ).

To prove the converse, let (X, τ ) be a topological space satisfying (2). Let apartτ

denote the corresponding apartness, and let apartτ (x, S) . Then x ∈ −S, where −S,being nearly open, is in τ (by Proposition 2.1). Applying (2), we obtain V ∈ τsuch that x ∈ V and X = −S∪ ∼ V. Let T =∼ V. Then X = −S ∪ T. Moreover,x ∈ V ⊂∼ T, so apartτ (x, T ) . �

P r o p o s i t i o n 2.3. A locally decomposable topological apartness space is topolog-ically consistent.

P r o o f. Let (X, τ ) be a locally decomposable topological apartness space. ByProposition 2.2, if U ∈ τ and x ∈ U, then there exists V ∈ τ such that x ∈ V andX = U∪ ∼V. Then x ∈ V ⊂∼∼ V, so apartτ (x,∼ V ). Since − ∼ V ⊂∼∼ V ⊂ U, itfollows that x ∈ − ∼ V ⊂ U. Thus U is a union of apartness complements and so isnearly open. �

It is easy to show that metric spaces (and uniform spaces, which we introduceshortly) are locally decomposable and hence topologically consistent.

Returning to the apartness topology, we say that a subset S of an apartness spaceX is nearly closed if

∀x ∈ X (near(x, S) ⇒ x ∈ S) .Both X and ∅ are nearly closed. The intersection of any family of nearly closed sets isnearly closed, but even in R we cannot expect any constructive proof that the unionof two nearly closed sets is nearly closed ([25], Section 6).

P r o p o s i t i o n 2.4. ([9], Proposition 27) If S is a nearly open subset of an apart-ness space X, then its logical complement equals its complement and is nearly closed.

We can characterise apartness and nearness topologically using the following com-bination of Propositions 28 and 29 of [9].

P r o p o s i t i o n 2.5. Let X be an apartness space. Then for each x ∈ X and eachA ⊂ X,

� apart (x, A) if and only if there exists B such that x ∈ −B ⊂∼A;� near(x, A) if and only if A intersects each nearly open subset of X that containsx.


It follows that if X is an apartness space, then its given apartness structure is thesame as that associated with the apartness topology. This observation is used in [9]to motivate the definition of the product of two apartness spaces X1 and X2.

There are at least three natural types of continuity for functions between apartnessspaces. We say that a mapping f : X → Y between apartness spaces is

• nearly continuous if ∀x ∈ X ∀A ⊂ X (near(x, A)⇒ near(f(x), f(A))) ;• continuous if ∀x ∈ X ∀A ⊂ X (apart(f(x), f(A)) ⇒ apart(x, A)) ;• topologically continuous if f−1(S) is nearly open in X for each nearly openS ⊂ Y.

P r o p o s i t i o n 2.6. ([9], Proposition 31) The following conditions are equivalenton a mapping f : X → Y between apartness spaces.

(i) f is nearly continuous.(ii) For each nearly closed subset T of Y, f−1(T ) is nearly closed.(iii) For each subset S of X, f(S) ⊂ f(S), where the closures are relative to the

apartness topologies on X, Y respectively.

For mappings between metric spaces, near continuity is equivalent to sequentialcontinuity, and apartness continuity to the constructively stronger notion of ε–δ con-tinuity [25].

Apartness continuity is a natural extension of the notion of strong extensionality:a function f : X → Y between sets with inequality relations is strongly extensionalif

∀x ∈ X∀x′ ∈ X (f(x) �= f(x′) ⇒ x �= x′) .In fact, strong extensionality holds even for nearly continuous functions.

P r o p o s i t i o n 2.7. A nearly continuous mapping between apartness spaces isstrongly extensional.

P r o o f. Let f : X → Y be nearly continuous and let x, y ∈ X be such thatf(x) �= f(y). Define

A = {z ∈ X : z = x ∨ (z = y ∧ x �= y)}.Note that x ∈ A. To show that near(y, A), consider any U ⊂ X such that y ∈ −U ;by axiom A5, either x �= y and therefore y ∈ A−U, or else x ∈ −U and so x ∈ A−U.Using the near continuity of f, we obtain near(f(y), f(A)). Since f(x) �= f(y), axiomA1 gives f(y) ∈ −{f(x)}; whence, by the definition of ‘near’, there exists z ∈ A suchthat f(z) �= f(x). Then ¬ (z = x) , so we must have z = y and x �= y. �

P r o p o s i t i o n 2.8. ([9], Proposition 32) A topologically continuous mapping be-tween apartness spaces is both continuous and nearly continuous.

Co r o l l a r y 2.9. Every topologically continuous mapping between apartness spacesis strongly extensional.

The following is a substantial improvement on Proposition 33 of [9], in that itreplaces a restrictive hypothesis (that of complete regularity) by a much simpler,more general one.


P r o p o s i t i o n 2.10. Let X and Y be apartness spaces, and assume that the fol-lowing condition holds in Y :

(3) apart (y, S) ⇒ ∃R ⊂ Y (apart (y, R) ∧ ¬R ⊂ −S).Then every continuous function f : X → Y is topologically continuous.

P r o o f. Let S ⊂ Y and x0 ∈ f−1(−S)—that is, f(x0) ∈ −S. It will suffice toconstruct T ⊂ X such that x0 ∈ −T and f(−T ) ⊂ −S. To do this, we use condition(3) to produce a subset R of Y such that f(x0) ∈ −R and ¬R ⊂ −S. DefiningT = f−1(R), we have f(T ) ⊂ R; whence

f(x0) ∈ −R ⊂∼R ⊂∼f(T )and therefore apart (f(x0), f(T )), by axiomA4. By the continuity of f , apart (x0, T ),so x0 ∈ −T. Now, if x ∈ −T, then x /∈ T—that is, f(x) /∈ R; whence f(x) ∈ ¬R ⊂ −S,as required. �

Co r o l l a r y 2.11. Every continuous mapping from an apartness space into a lo-cally decomposable apartness space is topologically continuous.

P r o o f. It is immediate that if Y is locally decomposable, then it satisfies (3), soProposition 2.10 applies. �

3 Set–set apartness

We now introduce a notion of apartness between subsets; the corresponding classicaltheory is sketched in [12] (Part II) and developed more fully in [18]. For more detailsof the constructive theory, see [8, 10, 11, 6, 15, 21, 20, 26].

As before, X will be a set. We also assume that there is a set–set apartnessrelation �� between pairs of subsets of X, such that the following axioms, in whichwe write

(4) −S = {x ∈ X : {x} �� S}hold.B1 X �� ∅B2 S �� T ⇒ S ∩ T = ∅B3 R �� (S ∪ T )⇔ R �� S ∧R �� TB4 {x} �� S ∧ −S ⊂∼ T ⇒ {x} �� TB5 {x} �� S ⇒ ∀y ∈ X (x �= y ∨ {y} �� S)B6 S �� T ⇒ T �� SB7 S �� T ⇒ ∀x ∈ X ∃R ⊂ X (x ∈ −R ∧ (∃y ∈ S −R ⇒ ¬R �� T ))

We then call X an apartness space, or, if clarity demands, a set–set apartnessspace.

Defining

x �= y ⇔ {x} �� {y}and

x �� S ⇔ {x} �� S,we obtain the inequality and the point–set apartness relation associated with thegiven set–set one, which satisfy the axioms A1–A5 under the additional assumption


that �= is nontrivial. Note that the corresponding apartness complement of a subsetS of X is precisely −S as defined by (4). Also, it readily follows from axioms B3 andB7 that the point–set apartness derived from �� satisfies condition (3) of Proposition2.10; so every apartness continuous mapping of a point–set apartness space into aset–set apartness space is topologically continuous.

It follows from B7 that the apartness topology on X is Hausdorff. (Indeed, ifx �= y in X—that is, {x} �� {y}, then, by B7, there is R ⊂ X with x ∈ −R and¬R �� {y}. By B6, y ∈ −¬R, and, by A2, −R ∩−¬R ⊂ ¬R ∩¬¬R, which is empty.)Note that our version of B7—i.e., that of [21]—is stronger than the one presented in[20]; the stronger version is needed for proofs in certain later papers, such as [10].

An example of a set–set apartness space is afforded by a certain type of uniformspace (X,U) , a type that includes metric spaces. It is shown in [20, 21] that therelation �� defined for subsets S, T of X by

S �� T ⇔ ∃U ∈ U (S × T ⊂∼ U)is a set–set apartness relation. The topology τU induced on X by U has an associatedrelation apartU defined by

apartU (x, S)⇔ ∃V ∈ τU (x ∈ V ⊂∼ S) .It turns out that, as we would hope, x �� S if and only if apartU (x, S).

The natural next step in set–set apartness theory is to investigate the analogueof uniform continuity. A mapping f between apartness spaces X, Y is said to bestrongly continuous if f (A) �� f (B) in Y implies that A �� B in X. Uniformcontinuity implies strong continuity. What about the converse?

Th e o r em 3.1. A strongly continuous mapping f : X → Y between metric spacesis uniformly sequentially continuous, in the sense that ρ (f(xn), f(x′n)) → 0whenever (xn) and (x′n) are sequences in X such that ρ(xn, x

′n) → 0.

This theorem is proved in [6]. There it is also shown that when the domain of f isseparable, the word ‘sequentially’ can be removed from the conclusion of Theorem 3.1provided that we are prepared to accept a certain additional hypothesis that holdsclassically and in the intuitionistic and recursive models of CM but does not appear tobe derivable in CM itself. Ishihara and Schuster [15] have shown that we can alsoremove ‘sequentially’ by assuming that the domain of f is totally bounded. Recently,Bridges and Vıta [11] have extended the Ishihara–Schuster result to uniform spaces:

Th e o r em 3.2. A strongly continuous mapping of a totally bounded uniformspace into a uniform space is uniformly continuous.

This extension suggests one approach to the tricky problem of introducing com-pactness into the theory. Note that compactness for a metric space means complete-ness plus total boundedness, since

� sequential compactness does not hold even for the discrete space {0, 1} ,11) and� the open–cover notion of compactness fails in the recursive model of CM ([7],

Chapter 3).

11)See, however, [5].


In view of Theorem 3.2, we can define an apartness space to be totally boundedif it is the image of a uniform space under a strongly continuous mapping. A com-pact apartness space is then an apartness space that is both totally bounded andcomplete. (For a discussion of Cauchy nets and completeness see [10].)

4 Regularity and uniformity

We say that a topological apartness space (X, τ ) is regular if the following conditionholds for all x and U :

(5) x ∈ U ∧ U ∈ τ ⇒ ∃V ∈ τ (x ∈ V ∧ ∀y ∈ X (y ∈ U ∨ y �� V )) ,where �� denotes the point–set apartness associated with τ. Clearly, regular implieslocally decomposable and hence topologically consistent, by Propositions 2.2 and 2.3.Note that conditions (2) and (5) differ from each other already in the classical theoryof metric spaces: although V ⊂ U in either condition, in the case of the former itmay happen that the boundaries of V and U have some point in common, a situationwhich is excluded in the case of the latter.

In this section we show how to extend the point–set apartness on a regular spaceto a set–set apartness. We then show that when this construction is applied tothe standard metric point–set apartness on R, the resulting set–set apartness is notinduced constructively by any uniform structure.

First, however, we ask: ‘What is the connection between our notion of regularityand the classical one?’ The latter notion is that if x does not belong to the closedset S, then there exist open sets V,W such that x ∈ V , S ⊂W, and V ∩W = ∅. Weshow that this is classically equivalent to our notion of regularity.

Classically, for an open set U we have x ∈ U if and only if x /∈∼ U, where ∼ U isclosed. Regularity now provides us with open sets V,W such that x ∈ V, ∼ U ⊂W,and V ∩W = ∅. For each y ∈ X we have (classically!) either y ∈ U or else y ∈∼U ⊂ W ⊂∼ V ; in the latter event, y �� V . Thus X = U ∪ −V, and we have verifiedregularity in our sense.

Again working classically, assume, conversely, that X is regular in our sense. Letx /∈ S, where S is closed in X. Then x ∈∼ S, which is open; so, by condition (5),there exists V ∈ τ such that x ∈ V and X = (∼ S)∪−V . Thus S is contained in theopen set −V , which is disjoint from V. We conclude that X is regular in the classicalsense. This completes the proof that our notion of regularity is classically equivalentto the classical one.

In the classical theory of proximity spaces [18], every T1 topological space (X, τ )has an associated set–set apartness given by

(6) S �� T ⇔ S ∩ T = ∅.This definition is not strong enough for us to verify axiom B7 even in the case X = R.However, as our next result shows, on a regular space we can define an apartness thatsatisfies our axioms and is classically equivalent to the one in (6).

P r o p o s i t i o n 4.1. Let (X, τ ) be a regular topological point–set apartness space,and for S, T ⊂ X define

(7) S �� T ⇔ ∀x ∈ X (x �� S ∨ x �� T ) ,


where, on the right–hand side, �� denotes the point–set apartness associated with τ .Then the set–set relation �� satisfies axioms B1–B7.

P r o o f. Since X is given to us as a topological point–set apartness space, we knowthat it satisfies axioms A1–A5; whence, by A2 and A5,

{x} �� T ⇔ x �� T

for all x ∈ X and T ⊂ X. In particular, X satisfies B4 and B5, which then coincidewith A4 and A5, respectively. For each x ∈ X we have x ∈ X ⊂∼ ∅ and thereforex �� ∅; from which B1 readily follows. If x ∈ S∩T, then x �� S∧x �� T, so ¬ (S �� T ) ;whence we obtain B2. Next,

R �� (S ∪ T ) ⇔ ∀x ∈ X (x �� R ∨ x �� (S ∪ T ))⇔ ∀x ∈ X (x �� R ∨ (x �� S ∧ x �� T )) by A3

⇔ ∀x ∈ X ((x �� R ∨ x �� S) ∧ (x �� R ∨ x �� T ))⇔ ∀x ∈ X ((x �� R ∨ x �� S)) ∧ ∀x ∈ X (x �� R ∨ x �� T )⇔ R �� S ∧R �� T,

so we have B3.Since B6 is clearly true, we are left to verify the awkward axiom, B7. To this end,

let S �� T and x ∈ X. Either x �� S or x �� T. In the first case we need only takeR = S in order to obtain

(8) x ∈ −R ∧ (S − R �= ∅ ⇒ ¬R �� T ) .In the case x �� T, choose U ∈ τ such that x ∈ U ⊂∼ T.

Using the regularity of X, we can find, in turn, V,W ∈ τ such that x ∈ V andX = U ∪ −V , and x ∈ W and X = V ∪ −W . Let R =∼W . Then

x ∈W ⊂∼∼W =∼ R,so x �� R. For each y ∈ X we have either y ∈ U ⊂∼ T and therefore y �� T , or elsey ∈ −V . In the latter case, noting that ¬R ⊂ ¬−W ⊂ V and therefore that

y ∈ −V ⊂∼ V ⊂∼ ¬R,we have y �� ¬R. Hence

∀y ∈ X (y �� ¬R ∨ y �� T )and therefore ¬R �� T. This completes the proof of B7 and hence that of the propo-sition. �

Note that in Proposition 4.1, the point–set apartness derived from the set–setapartness given by (7) is just the original point–set apartness on (X, τ ) .

We now consider the case X = R of Proposition 4.1. Denoting the standardmetric on R by ρ, we ask: is the apartness defined by

A �� B ⇔ ∀x ∈ R (ρ (x, A) > 0 ∨ ρ (x, B) > 0)

induced by a metric on R? Classically, the answer is ‘yes’, with the metric defined by

d (x, y) =∣∣∣∣x

1 + |x| −y

1 + |y|∣∣∣∣ .


The next proposition will enable us to show that there is no hope of proving con-structively that this apartness on R is even induced by a uniform structure.

P r o p o s i t i o n 4.2. If there exists an apartness space X such that(i) A �� B ⇒ ∀x ∈ X (x /∈ A ∨ x /∈ B) and(ii) any two disjoint subsets of a singleton are apart—that is, if x ∈ X,A ⊂

{x} , B ⊂ {x} , and A ∩B = ∅, then A �� B —then the law of excluded middle holds in the weak form

¬P ∨ ¬¬P.P r o o f. Suppose there exists such an apartness space X. Fixing ξ ∈ X, consider

any syntactically correct statement P, and define

A = {x : x = ξ ∧ P } ,B = {x : x = ξ ∧ ¬P } .

Then A and B are disjoint subsets of {ξ} , so, by (ii), A �� B. By hypothesis (i),either ξ /∈ A, in which case ¬P holds, or else ξ /∈ B and therefore ¬¬P holds. �

The somewhat eccentric hypothesis (ii) in the preceding proposition holds clas-sically for any apartness space: for if A,B are disjoint subsets of a singleton, thenclassically either A or B is empty, so axiom B1 applies. The hypothesis holds con-structively if the apartness on X is induced by a uniform structure U : for if ξ ∈ X,and A,B are disjoint subsets of {ξ}, then, taking any U ∈ U , we haveA×B = ∅ ⊂∼U ;whence A �� B.

Co r o l l a r y 4.3. Let ρ denote the standard metric on R, and let �� be the set–setapartness defined on R by

A �� B ⇔ ∀x ∈ R (ρ(x, A) > 0 ∨ ρ(x, B) > 0) .

If �� is induced by a uniform structure, then the law of excluded middle holds in theweak form

¬P ∨ ¬¬P.P r o o f. If �� is induced by a uniform structure, then any two disjoint subsets of

a singleton in X are apart, so Proposition 4.2 can be applied. �In view of Corollary 4.3, the constructive theory of apartness spaces is strictly

bigger than that of uniform spaces (which it certainly includes: see [21]). Note alsothat constructively the compact metric subspace [0, 1] of R has an apartness—thatgiven by Proposition 4.2—that induces the original topology but is not the sameas the original apartness; this contrasts with the classical situation in which anycompact topological space has a unique proximity structure compatible with theoriginal compact topology.

We end by examining the apartness in Corollary 4.3 under the additional hy-pothesis of Church’s thesis (in other words, working within recursive constructivemathematics as developed in [17]; see also Chapter 3 of [7]). We prove that thereexist nonempty, closed, bounded subsets S, T of R that are near relative to the stan-dard metric apartness, but apart with respect to the particular apartness �� definedas in Corollary 4.3. (This is impossible in classical mathematics. For, arguing classi-cally, suppose we have such S and T. Then we can find sequences (xn) in S and (yn)


in T such that ρ (xn, yn) → 0; since S is classically compact, we may assume thatxn → x∞ ∈ S. Then yn → x∞, so x∞ ∈ T = T and therefore x∞ ∈ S ∩ T , whichcontradicts the assumption that S �� T.)

Using Church’s thesis, we can produce Specker sequences: strictly increasing,bounded sequences whose terms are eventually bounded away from any given realnumber([7], Chapter 3). Let (sn) be a Specker sequence in the interval [0, 1] , and(tn) a (Specker) sequence with the following properties for each n (for instance, settn = sn +min{1/n, sn+1 − sn}/2 for each n):

• s1 < t1 < s2 < t2 < · · · ,• tn − sn < 1/n.

Let

S = {sn : n ≥ 1} , T = {tn : n ≥ 1} .Since |sn − tn| < 1/n for each n, we have ρ (S, T ) = 0. On the other hand, S �� T.To see the latter, let x ∈ R, and choose δ > 0 and a positive integer N such that|x− sn| ≥ δ for all n ≥ N. We may assume that δ > 2/N. Then |x− tn| ≥ 1/N forall n ≥ N. Either |x− sn| > 0 for all n < N, in which case ρ (x, S) > 0; or else thereexists k < N such that |x− sk| < ε, where

2ε = min{|sm − tn| : m, n < N} .In the latter case, |x− tn| > 0 for all n < N, and so ρ (x, T ) > 0 (because thealternative |x− t�| < ε for some # < N leads to the contradiction |sk − t�| < 2ε).

In retrospect, what we just have provided is a recursive counterexample to theimplication

S �� T ⇒ ρ(S, T ) > 0

for subsets of the unit interval, which we have seen before to be classically valid.(It is also intuitionistically valid when the subsets are compact.) That we cannotexpect a constructive proof of the converse either was clarified in [21] by means of aBrouwerian counterexample. More specifically, it was shown that the validity of thatconverse, for separable S, T ⊂ [0, 1], is equivalent to another weak form of the law ofexcluded middle, the Lesser Limited Principle of Omniscience:

For each binary sequence (an) with an = 1 for at most one n, eithera2n = 0 for all n or else a2n+1 = 0 for all n,

which also fails in recursive constructive analysis.

References

[1] Bishop, E., Foundations of Constructive Analysis. McGraw–Hill, New York 1967.[2] Bishop, E., Notes on Constructive Topology. University of California, San Diego 1971.[3] Bishop, E. and D. Bridges, Constructive Analysis. Grundlehren Math. Wiss. 279,

Springer–Verlag, Heidelberg–Berlin–New York 1985.[4] Bridges, D., Constructive mathematics: a foundation for computable analysis. Theo-

ret. Comput. Sci. 219 (1999), 95–109.[5] Bridges, D., H. Ishihara, and P. Schuster, Sequential compactness in constructive

analysis. Osterreich. Akad. Wiss. Math.–Natur. Kl. Sitzungsber. II 208 (1999) 159–163.


[6] Bridges, D., H. Ishihara, P. Schuster, and L. Vıta, Strong continuity impliesuniform sequential continuity. Preprint, University of Canterbury, Christchurch, NewZealand, 2001.

[7] Bridges, D. and F. Richman, Varieties of Constructive Mathematics. London Math.Soc. Lecture Notes 97, Cambridge University Press, Cambridge 1987.

[8] Bridges, D. and L. Vıta, Characterising near continuity constructively. Math. LogicQuart. 27(4) (2001), 535–538.

[9] Bridges, D. and L. Vıta, Apartness spaces as a framework for constructive topology.Ann. Pure Appl. Logic (to appear).

[10] Bridges, D. and L. Vıta, Cauchy nets in the constructive theory of apartness spaces.Sci. Math. Japon. 6 (2002), 123–132.

[11] Bridges, D. and L. Vıta, Strong continuity implies uniform continuity—the uniformspace case. Preprint, University of Canterbury, Christchurch, New Zealand, 2001.

[12] Cameron, P. J., J. G. Hocking, and S. A. Naimpally, Nearness—a better approachto topological continuity and limits (Parts I & II). Mathematics Report #18–73, Lake-head University, Canada, 1973.

[13] Curi, G., Metric spaces in type theory via formal topology. Electron. Notes Theoret.Comput. Sci. 35 (2000).

[14] Goodman N. D. and J. Myhill, Choice implies excluded middle. Zeit. Math. LogikGrundlagen Math. 24 (1978), 461.

[15] Ishihara H. and P. Schuster, A constructive uniform continuity theorem. Quart. J.Math. 53(2) (2002), 185–194.

[16] Kleene, S. C. and R. E. Vesley, The Foundations of Intuitionistic Mathematics.North–Holland, Amsterdam 1965.

[17] Kushner, B., Lectures on Constructive Mathematical Analysis. Amer. Math. Soc.,Providence RI 1985.

[18] Naimpally, S. A. and B. Warrack, Proximity Spaces. Cambridge Tracts in Math.and Math. Phys. 59, Cambridge University Press, Cambridge 1970.

[19] Sambin, G., Formal topology and domains. Electron. Notes Theor. Comp. Sci. 35(2000).

[20] Schuster, P., L. Vıta, and D. Bridges, Apartness as a relation between subsets.In: Combinatorics, Computability and Logic (Proceedings of DMTCS’01, Constanta,Romania, 2–6 July 2001; C.S. Calude, M.J. Dinneen, S. Sburlan, eds.), 203–214,DMTCS Series 17, Springer–Verlag, London 2001.

[21] Schuster, P., L. Vıta, and D. Bridges, Strong versus uniform continuity: a con-structive round. Preprint, University of Munich, Germany, 2001.

[22] Schuster, P., Unique existence, approximate solutions, and countable choice. Theoret.Comput. Sci. (to appear).

[23] Troelstra, A. S., Intuitionistic General Topology, Ph.D. Thesis, University of Ams-terdam 1966.

[24] Troelstra, A. S. and D. van Dalen, Constructivism in Mathematics: An Introduc-tion (two volumes). North Holland, Amsterdam 1988.

[25] Vıta, L. and D. Bridges, A constructive theory of point-set nearness. Theoret. Com-put. Sci. (to appear).

[26] Vıta, L., Proximal and uniform convergence on apartness spaces, Math. Logic. Quart.49 (2003).

[27] Waaldijk, F. A., Constructive Topology. Ph.D. thesis, University of Nijmegen 1996.[28] Weihrauch, K., Computable Analysis. Springer–Verlag, Heidelberg 2000.

(Received: December 14, 2001; Revised: April 23, 2002)

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Apartness, Topology, and Uniformity: a Constructive View