On metric spaces where continuous real valued functions are uniformly continuous in ZF

We show that the negation of each one of the following statements is consistent with ZF:(i) Every sequentially compact metric space X=(X,d)X=(X,d) is normal, i.e., the distance of any two disjoint non-empty closed subsets of X is strictly positive.(ii) If (X,d)(X,d) is a sequentially compact metric space then X is a UC space, i.e., every continuous real valued on X is uniformly continuous.(iii) If (X,d)(X,d) is a UC metric space then X is Lebesgue, i.e., every open cover of X has a Lebesgue number.(iv) If (X,d)(X,d) is a metric space such that every countable open cover of X has a Lebesgue number then X is Lebesgue.We also show:(v) For every metric space X, the following are equivalent:(1) Every sequence in X admits a Cauchy subsequence;(2) For every sequence (xn)n∈N(xn)n∈N of X, for each ε>0ε>0 there is an infinite Nε⊆NNε⊆N such that d(xn,xm)<εd(xn,xm)<ε whenever n,m∈Nεn,m∈Nε;(3) For every sequence (xn)n∈N(xn)n∈N of X, for each ε>0ε>0 and for each n0∈Nn0∈N there exist n,m∈Nn,m∈N, n,m≥n0n,m≥n0, n≠mn≠m such that d(xn,xm)<εd(xn,xm)<ε.(vi) The axiom of countable choice CAC implies that for every metric space X the following statements are equivalent:(1) X is Lebesgue;(2) Every countable open cover of X has a Lebesgue number;(3) X is UC.