Exercise 11, § 4

Deduce from Exercise 10 that:

a) If a product $\prod_{i \in I} X_i$ of non-empty Hausdorff spaces is normal, then $X_i$ is countably compact (§ 2, Exercise 14) for all save a countable set of indices.

b) A product $\prod_{i \in I} X_i$ of non-empty metrizable spaces is normal if and only if $X_i$ is compact for all but a countable set of indices; and the product space $\prod_{i \in I} X_i$ is then paracompact.