Exercise 29, § 4

Let $X$ be a Hausdorff topological space. Then $X$ is normal if and only if, for each closed subset $A$ of $X$ and each real-valued function $f$ defined on $A$ and continuous with respect to the induced topology, there exists a continuous pseudometric $d$ on $X$ with respect to which $f$ is uniformly continuous. (To show that the condition is sufficient, show that every real-valued function $f$ defined on a closed subset $A$, and continuous with respect to the induced topology, can be extended to a continuous function on $X$; for this, consider the Hausdorff space $\overline{X}$ associated with the uniformity defined on $X$ by the pseudometric $d$.)