Exercise 1, § 4

Let $(X_\alpha, f_{\alpha\beta})$ be an inverse system of topological spaces, whose indexing set is directed, $X = \varprojlim X$, and $f_\alpha$ be the canonical mapping from X to $X_\alpha$. Suppose that, for all $\alpha$, $f_\alpha$ is surjective; show that, if the $X_\alpha$ are irreducible, X is irreducible (cf. General Topology, Chapter I, § 4, no. 4, Corollary to Proposition 9). In particular, every product of irreducible spaces is irreducible.