LAUSR

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 .