A . Uniform sequential continuity, a property classically equivalent to uniform continuity on compact sets, is shown, constructively, to be a consequence of strong continuity on a metric space. It is then shown that in the case of a separable metric space, in order to omit the word sequential from this result, it is necessary and sufficient to adopt a principle (BD-N) that is independent of Hey...

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 continuou...