On metric spaces where continuous real valued functions are uniformly continuous in \(\mathbf{ZF}\) (Q306170)

In this paper, interrelations between compactness, completeness and open covering properties of metric spaces under the \textbf{ZF} axioms are studied. Relations, which are equivalent under the \textbf{ZFC} axioms, are placed in the deductive hierarchy of choice principles. In some ways the results are similar to the topics which are included in the book [\textit{P. Howard} and \textit{J. E. Rubin}, Consequences of the axiom of choice. Providence, RI: American Mathematical Society (1998; Zbl 0947.03001)]. In particular, independence results -- the negation of theorems in \textbf{ZFC} which are consistent with the \textbf{ZF} axioms -- are linked with models of the \textbf{ZF} axioms with non-trivial Dedekind-finite sets. All topics are consistently limited to the theory of metric spaces.