Proposition 10

Let $I$ be a directed set, let $(X_\alpha, f_{\alpha\beta})$ be an inverse system of uniform spaces indexed by $I$, and let $J$ be a cofinal subset of $I$. For each $\alpha \in I$ let $f_\alpha$ be the canonical mapping of $X = \varprojlim X_\alpha$ into $X_\alpha$, and let $g_\alpha$ denote $f_\alpha \times f_\alpha$. Then the family of sets $\overline{g}_\alpha^{-1}(V_\alpha)$, where $\alpha$ runs through $J$ and where, for each $\alpha \in J$, $V_\alpha$ runs through a fundamental system of entourages of $X_\alpha$, is a fundamental system of entourages of $X$.

We leave the proof to the reader; it is a straightforward adaptation of the proof of Proposition 9 of Chapter I, § 4, no. 4.

Finally, the topology on $X = \varprojlim X_\alpha$ induced by the inverse limit of the uniformities of the $X_\alpha$ is the inverse limit of the topologies of the $X_\alpha$.