Corollary

Let $X_1, X_2$ be two complete Hausdorff uniform spaces, and let $Y_1, Y_2$ be dense subspaces of $X_1, X_2$ respectively. Then every isomorphism $f$ of $Y_1$ onto $Y_2$ extends to an isomorphism of $X_1$ onto $X_2$.

$f$ is uniformly continuous in $Y_1$, hence (Theorem 2) extends to a uniformly continuous mapping $\bar{f} : X_1 \to X_2$. Likewise the inverse $g$ of $f$ extends to a uniformly continuous mapping $\bar{g} : X_2 \to X_1$. The function $\bar{g} \circ \bar{f}$ is therefore a continuous mapping of $X_1$ into itself whose restriction to $Y_1$ is the identity mapping; by the principle of extension of identities (Chapter I, $\S$ 8, no. 1, Corollary 1 to Proposition 2) $\bar{g} \circ \bar{f}$ is therefore the identity mapping of $X_1$; similarly $\bar{f} \circ \bar{g}$ is the identity map of $X_2$. Consequently (Set Theory, R, $\S$ 2, no. 12) $\bar{f}^{-1}$ and $\bar{g}$ are bijections and are inverses of each other; they are also uniformly continuous and are therefore isomorphisms ($\S$ 2, no. 1, Proposition 2).

It should be remarked that if $f$ is a bijective uniformly continuous mapping of $Y_1$ onto $Y_2$, its extension by continuity need be neither injective nor surjective (Exercise 3).