On metric spaces where continuous real valued functions are uniformly continuous and related notions

Given a metric space X = (X, d) we show in ZF that: (a) The following are equivalent: (i) For every two closed and disjoint subsets A, B of X, d(A, B) > 0. (ii) Every countable open cover of X has a Lebesgue number. (iii) Every real valued continuous function on X is uniformly continuous. (iv) For every countable (resp. finite, binary) open cover U of X, there exists a delta > 0 such that for all x, y is an element of X with d(x, y) < delta, {x, y} subset of U for some U is an element of U. (b) If X is connected then: X is countably compact iff every open cover of X has a Lebesgue number iff for every two closed and disjoint subsets A, B of X, d(A, B) > 0. (C) 2018 Elsevier B.V. All rights reserved.