Corollary 1

Let $X$ be a uniform space and let $i$ be the canonical mapping of $X$ into its Hausdorff completion $\hat{X}$; let $A$ be a subspace of $X$ and $j : A \to X$ the canonical injection. Then $\hat{j} : \hat{A} \to \hat{X}$ is an isomorphism of $\hat{A}$ onto the closure of $i(A)$ in $\hat{X}$.