Corollary 2

*Let $(X_\alpha, f_{\alpha \beta})$ be an inverse system of topological spaces. If the $X_\alpha$ are Hausdorff, then $X = \varprojlim X_\alpha$ is Hausdorff and is a closed subspace of $\prod_\alpha X_\alpha$.

The first assertion follows from the fact that $X$ is a subspace of the Hausdorff space $\prod_\alpha X_\alpha$ (Proposition 7). To show that $X$ is closed in the product space, let $F_{\alpha \beta}$ ($\alpha \leq \beta$) be the subset of $\prod_\alpha X_\alpha$ consisting of points $x$ for which $\operatorname{pr}_\alpha x = f_{\alpha \beta}(\operatorname{pr}_\beta x)$; the $F_{\alpha \beta}$ are closed in $\prod_\alpha X_\alpha$ (no. 1, Proposition 2), hence so is their intersection $X$.

Evidently, any sum of Hausdorff spaces (§ 2, no. 4, Example 3) is a Hausdorff space.