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Uniform continuity and normality of metric spaces in $\mathbf{ZF}$

2017
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Bulletin of the Polish Academy of Sciences Mathematics
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Let X = (X, d) and Y = (Y, ρ) be two metric spaces. (a) We show in ZF that: (i) If X is separable and f : X → Y is a continuous function then f is uniformly continuous iff for any A, B ⊆ X with d(A, B) = 0, ρ(f (A), f (B)) = 0. But it is relatively consistent with ZF that there exist metric spaces X, Y and a continuous, nonuniformly continuous function f : (ii) If S is a dense subset of X, Y is Cantor complete and f : S → Y a uniformly continuous function, then there is a unique uniformly

doi:10.4064/ba8122-10-2017
fatcat:ui3il6nipvb77iwlrblfhoqhse