Exercise 3, § 4

Let $f$ be a continuous mapping of a compact space $X$ into a compact space $Y$, and let $V$ be an open entourage of $X$ such that the relation $f(x) = f(y)$ implies $(x, y) \in V$. Show that there is an entourage $W$ of $Y$ such that the relation $(f(x), f(y)) \in W$ implies $(x, y) \in V$.