Math. Log. Quart. 47 (2001) 4, 535 – 538

Mathematical LogicQuarterly

c© WILEY-VCH Verlag Berlin GmbH 2001

Characterising Near Continuity Constructively

Douglas Bridges and Luminita Vıta1)

Department of Mathematics and Statistics, University of Canterbury,Private Bag 4800, Christchurch, New Zealand2)

Abstract. The relation between near continuity and sequential continuity for mappingsbetween metric spaces is explored constructively. It is also shown that the classical impli-cations “near continuity implies sequential continuity” and “near continuity implies apartcontinuity” are essentially nonconstructive.

Mathematics Subject Classification: 03F60, 54A05.

Keywords: Near continuity, Sequential continuity, Apartness, Constructive Analysis.

This note arose out of a project in which the authors, with others, are developingconstructive3) topology based on notions of nearness and apartness between pointsand sets and between sets and sets (see [11, 3, 9]). In the context of a metric space(X, �), we say that a point x is near a set S if (∀ε > 0) (∃s ∈ S)(�(x, s) < ε), and wesay that x is apart from S if (∃r > 0) (∀s ∈ S) (�(x, s) ≥ r). In the first case we writenear(x, S), and in the second apart(x, S). With classical logic we have

near(x, S) ⇔ ¬apart(x, S).

However, our constructive theory is based on intuitionistic logic, with which we cannotderive the implication from right to left. Thus in our context we have several (clas-sically equivalent) notions of continuity of a mapping f : X −→ Y between metricspaces:

· f is nearly continuous at x, in which near(x, S) implies near(f(x), f(S));· f is apart continuous at x, in which apart(f(x), f(S)) implies apart(x, S);· f is sequentially continuous at x, in the sense that f(xn) converge to f(x) when-

ever (xn) is a sequence converging to x;· f is pointwise continuous at x with the usual ε–δ definition.

1)The authors are grateful to the referee for suggesting simplifications of their original proofs ofPropositions 1 and 2

2)e-mail: d.bridges@math.canterbury.ac.nz3)We should make it clear that we use “constructive mathematics” to denote Bishop’s constructive

mathematics, which is, in practice, mathematics with intuitionistic logic. Two standard models ofour constructive mathematics are Brouwer’s intuitionistic mathematics (see [4]) and the recursiveconstructive mathematics of the school of Markov (see [8]) . However, all our proofs and results holdin classical mathematics – that is, mathematics with classical logic. For background on constructivemathematics see [1, 2, 10].

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536 Douglas Bridges and Luminita Vıta

Pointwise continuity at x implies sequential continuity there; but in order to provethe converse, we need a principle (called BD-N by Ishihara [7]) which does not appearprovable solely with intuitionistic logic, even though it holds classically, intuitionisti-cally, and in recursive constructive mathematics. We proved in [11] that f is apartlycontinuous at x if and only if it is pointwise continuous there; that if f is sequentiallycontinuous at x, then it is nearly continuous at x; and that if X is complete and f isboth nearly continuous at x and strongly extensional4), in the sense that f(s) = f(t)– that is, �(f(s), f(t)) > 0) – entails s = t, then f is sequentially continuous at x.In the present note we first show that a nearly continuous mapping between metricspaces is always strongly extensional. We then provide a necessary and sufficientcondition for near continuity in the absence of completeness. Finally, we show, by aBrouwerian example, that even in the intuitionistic or recursive models of construc-tive mathematics, in the absence of completeness of the domain we cannot prove thatnear continuity implies sequential continuity, let alone apart continuity.

P r o po s i t i o n 1. A nearly continuous mapping between metric spaces is stronglyextensional.

P r o o f . Let f : X −→ Y be a nearly continuous mapping between metric spaces,and let x, y be points of X such that f(x) = f(y). Construct an increasing binarysequence (λn) such that

λn = 0 ⇒ �(x, y) < 1/n, λn = 1 ⇒ �(x, y) > 0.

We may assume that λ1 = 0. If λn = 0, set An = {x}; if λn = 1, set An = {y}.Then near(y, A), where A =

⋃∞n=1 An. For if n is a positive integer, then either

λn = 0 and �(y, x) < 1/n, where x ∈ A; or else λn = 1 and y ∈ A. Since f isnearly continuous, near(f(y), f(A)). Thus there exists N and an element a of AN

such that �(f(y), f(a)) < �(f(y), f(x)). It follows that ¬(a = x); whence ¬(λN = 0)and therefore λN = 1. ✷

P r o po s i t i o n 2. Let f : X −→ Y be a nearly continuous mapping between metricspaces, and let (xn) be a sequence in X that converges to a point x ∈ X. Then for eachpositive integer m and each ε > 0 there exists n ≥ m such that �(f(xn), f(x)) < ε.

P r o o f . Given m, let A = {xk : k ≥ m}. Then near(x, A), so near(f(x), f(A))and therefore for each ε > 0 there exists xn ∈ A such that �(f(xn), f(x)) < ε. ✷

Th e o r em 3. Let f : X −→ Y be a mapping between metric spaces, and letx ∈ X. Then the following conditions are equivalent:

(i) f is nearly continuous at x.(ii) If (xn) is a sequence converging to x, then f(x) is a cluster point of the sequence(f(xn))∞n=1.

If f is nearly continuous at x, then f(x) is the unique cluster point of the sequence(f(xn))∞n=1.

4)In constructive mathematics, strong extensionality is a stronger property than functionality.In order to prove that every function between metric spaces is strongly extensional, we need toassume Markov’s Principle, a form of unbounded search that is not used by most constructivemathematicians and that is independent of intuitionistic logic (see [4, p. 137]). Note that withoutMarkov’s Principle, although, for real numbers x and y, we can establish that (¬(x �= y) ⇒ x = y),we cannot prove that (¬(x = y) ⇒ x �= y).

Characterising Near Continuity Constructively 537

P r o o f . Suppose first that f is nearly continuous at x and let (xn) be a sequencein X that converges to x. Setting n0 = 0 and applying Proposition 2 inductivelyto the sequence (xn)∞n=nk

, we construct a strictly increasing sequence (nk)∞k=0 suchthat �(f(xnk ), f(x)) < 1/k for each k. Thus (f(xn))∞n=1, and similarly each of itssubsequences, has f(x) as a cluster point. It follows that f(x) is the unique clusterpoint of (f(xn))∞n=1.

Now assume (ii), and let x ∈ X and S ⊂ X satisfy near(x, S). Choose a sequence(xn) in S converging to x. By (ii), there is a subsequence (xnk)∞k=1 such that f(xnk )converges to f(x); whence near(f(x), f(S)). Thus (ii)⇒ (i). ✷

Co r o l l a r y 4. A mapping f : X −→ Y between metric spaces is nearly continuousif and only if it preserves cluster points: that is, for each cluster point x of a sequence(xn) in X, f(x) is a cluster point of the sequence (f(xn)) in Y .

P r o o f . This is a simple consequence of Theorem 3. ✷

Co r o l l a r y 5. A nearly continuous mapping f : X −→ Y between metric spacesis sequentially nondiscontinuous, that is, if (xn) is a sequence converging to x in Xand if �(f(xn), f(x)) ≥ α for all n, then α ≤ 0.

P r o o f . If f : X −→ Y is such a mapping, �(xn, x)→ 0, and �(f(xn), f(x)) ≥ αfor each n, then as, by Theorem 3, f(x) is a cluster point of (f(xn))∞n=1, there existsn such that �(f(xn), f(x)) is arbitrarily close to 0; whence α ≤ 0. ✷

Using Proposition 1, Theorem 3, and a result of Ishihara [6], we see that if amapping from a complete metric space into a metric space is nearly continuous at apoint x, then it is sequentially continuous at x (cf. [11, Proposition 1]). The questionremains: can we remove the completeness hypothesis from this last remark ? Thefollowing Brouwerian example shows that we cannot.

Consider the metric subspace

X = {n−1 : n = 1, 2, 3, . . .} ∪ {0}of R. Given a binary sequence (an)∞n=1 with at most one term equal to 1, define amapping f : X −→ {0, 1} by setting f(n−1) = an and f(0) = 0. To show that fis nearly continuous, let x ∈ X, S ⊂ X, and near(x, S). Either x = n−1 for someinteger n – in which case x ∈ S and therefore, trivially, near(f(x), f(S)) – or else x = 0.In the latter case, choose any y ∈ S. If y = 0, then we are again in the case x ∈ S.If y = N−1 for some integer N , then either aN = 0 – so f(x) = 0 = f(N−1) ∈ f(S)and therefore near(f(x), f(S)) – or else aN = 1. In that case, since near(0, S), we canfind z ∈ S such that z < y; if z = 0, then we are done as before; if z = m−1 for someinteger m, then f(z) = am = 0 = f(0) and again we are done.

If, however, f is sequentially continuous at 0, then an = f(n−1) converges tof(0) = 0. Choosing N such that f(n−1) < 1 for all n ≥ N , we can test the termsa1, . . . , aN−1 to check whether or not any of them equals 1; if they are all 0, thenan = 0 for all n. Thus we see that the statement

“Every nearly continuous mapping between metric spaces is sequentiallycontinuous”

entails the limited principle of omniscience (LPO),“For each binary sequence (an) either an = 0 for all n or else thereexists n such that an = 1”.

538 Douglas Bridges and Luminita Vıta

The latter is known to be essentially nonconstructive; indeed it is false in both theintuitionistic and recursive models of constructive mathematics (see [2]).

The foregoing Brouwerian example also confirms to show that we cannot proveconstructively that near continuity implies apart continuity: if A = {n−1 : an = 1},we see that |f(0) − f(x)| = 1 for each x ∈ A; but if there exists r > 0 such that|0−x| ≥ r for each x ∈ A, then by testing ak for each k > r−1, we can decide whetheror not there exists n with an = 1.

References

[1] Bishop, E., Foundations of Constructive Analysis. McGraw-Hill, New York 1967.

[2] Bishop, E., and D. Bridges, Constructive Analysis. Springer-Verlag, Berlin-Heidel-berg-New York 1985.

[3] Bridges, D., and L. Vı ta, Apartness as a framework for constructive topology.Preprint, University of Canterbury, New Zealand, 2000.

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

[5] Dummett, M., Intuitionism: An Introduction. Second Edition. Oxford Univ. Press,Oxford 2000.

[6] Ishihara, H., Continuity and nondiscontinuity in constructive mathematics. J. Sym-bolic Logic 56 (1991), 1349 – 1354.

[7] Ishihara, H., Continuity properties in constructive mathematics. J. Symbolic Logic 57(1992), 557 – 565.

[8] Kushner, B. A., Lectures on Constructive Mathematical Analysis. Amer. Math. Soc.,Providence (RI) 1985.

[9] Schuster, P., D. Bridges, and L. Vı ta Dediu, Apartness as a relation between sub-sets. Preprint, University of Canterbury, New Zealand, 2000.

[10] Troelstra, A. S., and D. van Dalen, Constructivism in Mathematics: An Introduc-tion. North-Holland Publ. Comp., Amsterdam 1988.

[11] Vı ta, L., and D. Bridges, A first-order constructive theory of nearness spaces.Preprint, University of Canterbury, New Zealand, 2000.

(Received: September 15, 2000; Revised: February 7, 2001)