Corollary 2

Let $(X_\alpha, f_{\alpha\beta})$ and $(X'_\alpha, f'_{\alpha\beta})$ be two inverse systems of topological spaces indexed by the same directed set $I$, and let $(u_\alpha)$ be an inverse system of mappings $u_\alpha : X_\alpha \to X'_\alpha$. Let $X = \varprojlim X_\alpha$, $X' = \varprojlim X'_\alpha$, $u = \varprojlim u_\alpha$. Then: a) If $x' = (x'_\alpha) \in X'$ is such that $\overline{u'}_{\alpha}^{-1}(x'_\alpha)$ is compact and non-empty for each $\alpha \in I$, then $\overline{u}^{-1}(x')$ is compact and non-empty.

b) If the $X_\alpha$ are compact, the $X'_\alpha$ Hausdorff and the $u_\alpha$ surjective and continuous, then $u$ is surjective.

Let $L_\alpha$ denote $\overline{u}_\alpha^{-1}(x'_\alpha)$; then clearly the $L_\alpha$ form an inverse system of topological spaces (with the restrictions of the $f_{\alpha\beta}$ as maps) and $\overline{u}^{-1}(x' = L)$ is the inverse limit of the $L_\alpha$; assertion a) therefore follows from Proposition 8. Assertion b) is an immediate consequence, in view of Proposition 3 of no. 3.