Exercise 17, § 5

a) Show that every product $X = \prod_{i \in I} X_i$ of complete metric spaces is a Baire space. [Argue as in the proof of Theorem 1, taking $G_n$ to be an elementary set (Chapter I, § 4, no. 1) all of whose projections are either the whole space or have a diameter $\leq 1/n$; then note that there is a countable subset $J$ of $I$ such that each $G_n$ is of the form $H_n \times \prod_{i \notin J} X_i$, where $H_n \subset \prod_{i \in J} X_i$.]

b) Show by an example that if $I$ is uncountable then $X$ need not be totally non-meagre (Exercise 12). [Use Exercises 1 and 23 b) of § 2.]