A Metric Extension Theorem

Abstract

We show that, given a partially defined metric $d:R\rightarrow\mathbb{R}$ where $R\subset X^2$ is a closed equivalence relation on a compact metrizable space $X$, there is a metric $\hat{d}:X^2\rightarrow \mathbb{R}$ such that $d = \hat{d}|_{R}$