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

Kyriakos Keremedis
2017 Bulletin of the Polish Academy of Sciences Mathematics
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
more » ... uous function F : X → Y extending f . But it is relatively consistent with ZF that there exist a metric space X, a complete metric space Y, a dense subset S of X and a uniformly continuous function f : S → Y that does not extend to a uniformly continuous function on X. (iii) X is complete iff for any Cauchy sequences (xn) n∈N and (yn) n∈N in X, if {xn : n ∈ N} ∩ {yn : n ∈ N} = ∅ then d({xn : n ∈ N}, {yn : n ∈ N}) > 0. (b) We show in ZF+CAC that if 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. 2010 Mathematics Subject Classification: 03E25, 54E35, 54E45, 54E50, 54C20, 54C30.