Exercise 3, § 10

Let $(X_\alpha, f_{\alpha\beta})$ be an inverse system of quasi-compact Kolmogoroff spaces such that the $f_{\alpha\beta}$ are proper mappings. Show that if the $X_\alpha$ are non-empty then $\varprojlim X_\alpha$ is not empty (show that in a quasi-compact Kolmogoroff space, every non-empty closed subset contains a closed subset consisting of a single point).