Strong Proximal Continuity and Convergence

and Applied Analysis 3 that the function f : (X, α) → (Y, β) is strongly proximally continuous on A if ∀E ⊂ A, ∀S ⊂ X, EαS 󳨐⇒ f (E) βf (S) . (5) Finally, one says that f is strongly proximally continuous on B if f is strongly proximally continuous on A, for every A ∈ B. We shall use the notation C B (X, Y) to denote the family of the functions from X to Y which are strongly proximally continuous on the bornologyB. Remark 8. Proximal and strong proximal continuity on a set can be equivalently expressed in the following way: f is proximally continuous on A if ∀E ⊂ A, ∀T ⊂ f (A) , f (E)≪ β T 󳨐⇒ E≪ α f −1 (T) . (6) f is strongly proximally continuous on A if ∀E ⊂ A, ∀T ⊂ Y, f (E)≪ β T 󳨐⇒ E≪ α f −1 (T) . (7) We now recall some connections among the introduced continuity notions. Strong uniform continuity on a set implies uniform continuity on that set. The converse does not hold in general; for a complete characterization of equivalence among the two continuities in uniform spaces, see Beer [5]; strong proximal continuity implies proximal continuity, and in general the two notions do not coincide; furthermore (strong) uniform continuity implies (strong) proximal continuity, for the natural proximity associated to the uniformity. On the contrary a proximally continuous function needs not to be uniformly continuous for the uniformities compatible with the proximity. As an example, take a proximity space (X, α) and uniformities U 2 and U 1 on X such thatU 1 is strictly finer thanU 2 . The identity map Id : (X,U 2 ) → (X,U 1 ) is not uniformly continuous, but the associated map from the proximal space (X, α) into itself is so. However, if X, Y are metric spaces and f is proximally continuous with respect to the induced proximities, then it is also uniformly continuous with respect to their underlying uniformities (see [17, Corollary 4.4.2]). More generally, it is clear that on the whole bornology B f the two notions of strong continuity do coincide and are equivalent to continuity of f. For the coincidence of the two notions of uniform continuity and strong uniform continuity on a bornology, let us recall the result proved in Beer [5, Theorem 3.5] that a continuous function acting between two uniform spaces, if it is uniformly continuous on a bornology shielded from closed sets, is automatically strongly uniformly continuous on the bornology. To provide a similar result in the proximity setting, we rephrase the notion of bornology stable under small enlargements, given in Beer [5, Definition 3.2] in a uniform setting, in order to have the analogous property in proximal spaces. Definition 9. Given a proximity space (X, α) and a bornology B on it, one says thatB is stable under small enlargements if for every A inB there isB inB such that A≪ α B. The next Lemma is useful to prove when proximal continuity and strong proximal continuity coincide on a bornology. Lemma 10. Let (X, α) and (Y, β) be proximity spaces. Let A,C ⊂ X such that A≪ α C. Suppose that for each F ⊂ C such that AαF it holds f(A)βf(F). Then f is strongly proximally continuous on A. Proof. Take G ⊂ X such that AαG. Let F = C ∩ G. We claim thatAαF. Otherwise, settingH = G∩C, it isAαH, but then G = F ∪ H is such that AαG, against the assumption. Thus AαF and then by assumption f(A)βf(F). Since F ⊂ G, it follows that f(A)βf(G). Proposition 11. Let (X, α) and (Y, β) be two proximity spaces. Let B be a bornology on X stable under small enlargements. Then the following are equivalent: (1) f is proximally continuous onB; (2) f is strongly proximally continuous onB. Proof. We only need to prove that (1) implies (2). Fix A ∈ B and let C ∈ B be such that A≪ α C. From Lemma 10 it is enough to prove that if F ⊂ C is such that AαF, then f(A)βf(F). But this is an immediate consequence of the fact that f is proximally continuous on C. Since the bornologyB f is not stable under small enlargements, but it is shielded from closed sets, it is easily seen that in the above proposition the assumption on the bornology of being stable under small enlargements cannot be weakened, since any function is proximally continuous onB f , but only the continuous functions are strongly proximally continuous onB f . In Beer [5, Theorem 3.5] a result similar to Proposition 11 in uniform setting is proved.