Corollary

Let $(X_\alpha, f_{\alpha\beta})$ be an inverse system of uniform spaces. If the $X_\alpha$ are Hausdorff and complete, then so is $X = \varprojlim X_\alpha$.

For $X$ is Hausdorff and can be identified with a closed subspace of $\prod_{i \in I} X_\alpha$ (Chapter I, § 8, no. 2, Corollary 2 to Proposition 7); the corollary therefore follows from Proposition 10 and Proposition 8 (no. 4).

An inverse limit of complete Hausdorff uniform spaces $X_\alpha$ can be empty, even if all the $X_\alpha$ are non-empty and all the $f_{\alpha\beta}$ are surjective, as is shown by the case of discrete spaces (Set Theory, Chapter III, § 1, Exercise 32). However, we have the following theorem: