Corollary 1

Let $(X_\alpha, f_{\alpha\beta})$ be an inverse system of topological spaces indexed by a directed set, such that for each pair of indices $\alpha, \beta$ for which $\alpha \leq \beta$ and for each $x_\alpha \in X_\alpha$, $\overline{f}_{\alpha\beta}^{-1}(x_\alpha)$ is compact. Then equation (1) is valid and $\overline{f}_{\alpha}^{-1}(x_\alpha)$ is compact for each index $\alpha$ and each $x_\alpha \in X_\alpha$.

For each $x_\alpha \in \bigcap_{\beta \geq \alpha} f_{\alpha\beta}(X_\beta)$ and each $\beta \geq \alpha$, let $L_\beta$ denote $\overline{f}_{\alpha\beta}^{-1}(x_\alpha)$. If $\alpha \leq \beta \leq \gamma$, then we have $f_{\beta\gamma}(L_\gamma) \subset L_\beta$ and the set of all indices $\beta \geq \alpha$ is cofinal in the index set. It follows immediately that the $L_\beta$ ($\beta \geq \alpha$) form an inverse system of topological spaces (with the restrictions of the $f_{\beta\gamma}$ as mappings), whose inverse limit $L$ is homeomorphic to $\overline{f}_{\alpha}^{-1}(x_\alpha)$. Since by hypothesis the $L_\beta$ are compact and not empty, the corollary follows from Proposition 8.