Exercise 22, § 1

Let $X$ be a Hausdorff uniform space, let $Y$ be a closed subset of $X$, and let $f$ be a bounded, uniformly continuous, real-valued function on $Y$. Show that $f$ has an extension $\overline{f}$ to $X$ which is bounded and uniformly continuous on $X$. [We may assume that $f(Y) \subset [0, 1]$. For each dyadic number $r = k/2^n \in [0, 1]$, let $A(r)$ be the set of all $x \in Y$ such that $f(x) \leq r$, and let $B(r) = A(r) \cup (X - Y)$. Define by induction \* (as in the proof of Theorem 1 of § 4, no. 1) \* an open set $U(r)$ for each dyadic number $r \in [0, 1]$ such that, whenever $r$ and $r'$ are two dyadic numbers and $r < r'$, there is an entourage $V$ of the uniformity for $X$ for which

Then define $\overline{f}(x)$ to be the greatest lower bound of the dyadic numbers $r$ such that $x \in U(r)$.]