Exercise 6, § 3

Let $X$ be a uniform space and let $i : X \to \hat{X}$ be the canonical mapping of $X$ into its Hausdorff completion. Let $X_0$ be the sum of $X$ and $\hat{X} - i(X)$, and let $j : X_0 \to \hat{X}$ be the mapping which agrees with $i$ on $X$ and with the identity mapping on $\hat{X} - i(X)$. Show that $X_0$ is complete with respect to the uniformity which is the inverse image under $j$ of the uniformity of $\hat{X}$, also that if $Y$ is any complete uniform space such that $X$ is a uniform subspace of $Y$, then the identity mapping $X \to X$ can be extended uniquely to a continuous mapping $X_0 \to Y$.