Exercise 7, § 9

a) Give an example of an inverse system $(X_\alpha, f_{\alpha \beta})$ of quasi-compact spaces whose inverse limit is empty (observe that in the space of Example 2 of no. 1, every subspace is quasi-compact).

b) Let $p$ be a prime number. Let $Z_n$ denote the set $\mathbf{Z}$ of rational integers, topologized with the inverse image of the discrete topology under the canonical mapping of $\mathbf{Z}$ onto $\mathbf{Z}/p^n \mathbf{Z}$. If $m \leq n$, let $f_{mn}$ denote the identity mapping $Z_n \to Z_m$, which is continuous. Show that the spaces $Z_n$ are quasi-compact, but that the inverse limit of the inverse system $(Z_n, f_{mn})$ is not quasi-compact (*).