Metric spaces and the axiom of choice

We shall start with some definitions from topology. First of all, a metric space is a topological space whose topology is determined by a metric. A metric on a topological space X is a function d from X × X to R , the reals, which has the following properties: For all x, y, z ∈ X , (a) d(x, y) ≥ 0, (b) d(x, x) = 0, (c) if d(x, y) = 0, then x = y, (d) d(x, y) = d(y, x), and (e) d(x, y) + d(y, z) ≥ d(x, z). Functions which satisfy (a), (b), (d), and (e) are called pseudometrics. We let “AC” be the axiom of choice, “ZF” Zermelo-Fraenkel set theory, “ZF”, ZF without the axiom of foundation and “ZFC”, ZF +AC. Additional definitions follow.