Exercise 10, § 2

Let $X$ be a compact space. If there exists a continuous real-valued function $f$ on $X \times X$ such that the relation $f(x, y) = c$ is equivalent to $x = y$, then $X$ is metrizable. (Show that if $V$ runs through a fundamental system of neighbourhoods of $o$ in $\mathbf{R}$, the sets $f^{-1}(V)$ form a fundamental system of neighbourhoods of the diagonal $\Delta$ in $X \times X$.)