Proposition 8

Let $(X_\alpha, f_{\alpha\beta})$ be an inverse system of compact spaces indexed by a directed set $I$ such that $f_{\alpha\alpha}$ is the identity mapping for each $\alpha \in I$. Let $X = \varprojlim X_\alpha$ be the inverse limit and $f_\alpha : X \to X_\alpha$ the canonical mapping (§ 4, no. 4). Then a) $X$ is compact and for each $\alpha \in I$ we have

b) If the $X_\alpha$ are all non-empty then $X$ is non-empty.

$X$ is a closed subspace of $\prod_\alpha X_\alpha$ (§ 8, no. 2, Proposition 7, Corollary 2) which is compact by Theorem 3 of no. 5 and Proposition 3 of no. 3. The other assertions are consequences of Set Theory, chap. III, § 7, no. 4, th. 1. We apply this theorem by taking $\mathscr{S}_\alpha$ to be the set of closed subsets of $X_\alpha$. The conditions (i) and (ii) are just axioms $(O'_1)$ and $(C'')$ respectively; condition (iii) is satisfied since $\{x_\alpha\}$ is closed and $f_{\alpha\beta}$ continuous (§ 2, no. 1, Theorem 1), and lastly condition (iv) is satisfied by reason of Corollary 2 of Theorem 2 of no. 4.