On uniformly continuous functions between pseudometric spaces and the Axiom of Countable Choice

Abstract

In this note we show that the Axiom of Countable Choice is equivalent to two statements from the theory of pseudometric spaces: the first of them is a well-known characterization of uniform continuity for functions between (pseudo)metric spaces, and the second declares that sequentially compact pseudometric spaces are $$\mathbf {UC}$$ —meaning that all real valued, continuous functions defined on these spaces are necessarily uniformly continuous.