Abstract
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 δ>0 such that for all x,y∈X with d(x,y)<δ, {x,y}⊆U for some U∈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.
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