Exercise 14, § 4

, but the product $X \times Y$ is not normal (use Exercise 12 of I, p. 106).

¶ 13) Let L be an uncountable set; consider the completely regular space $X = \mathbf{N}^L$ ($\mathbf{N}$ being endowed with the discrete topology). Let A (resp. B) be the set of the $x = (x(\lambda))_{\lambda \in L}$ in X such that, for every integer $k \neq 0$ (resp. $k \neq 1$), the set of the $\lambda \in L$ such that $x(\lambda) = k$ has at most one element. (a) Prove that A and B are closed and have no point in common in X. (b) Let U, V be two open sets in X such that $A \subset U, B \subset V$. Denote by $\Phi$ the set of elementary sets (I, p. 24) in X whose projections distinct from $\mathbf{N}$ are reduced to a single element; for every $W \in \Phi$, let $H(W)$ be the set of the $\lambda \in L$ such that $\operatorname{pr}_\lambda(W)$ is reduced to a single element. Prove that there exists a sequence $(x_n)$ of points of A, a sequence $(U_n)$ of sets of $\Phi$, a sequence $(\lambda_n)$ of distinct elements of L and a strictly increasing sequence $(m(n))_{n \in \mathbf{N}}$ of integers, having the following properties: $1^\circ U_n \subset U$ is a neighbourhood of $x_n$; $2^\circ H(U_n)$ is the set of the $\lambda_k$ such that $k \leq m(n)$; $3^\circ$ one has $x_0(\lambda) = 0$ for all $\lambda \in L$ and for all $n > 0$, $x_n(\lambda_k) = k$ if $k \leq m(n - 1)$ and $x_n(\lambda) = 0$ for the other $\lambda \in L$.

(c) Let $y \in B$ be the point such that $y(\lambda_k) = k$ for every integer $k$ and $y(\lambda) = 1$ for the $\lambda \in L$ distinct from the $\lambda_k$. Let $V_0 \subset V$ be a set of $\Phi$ containing $y$ and let $n$ be an integer such that $\lambda_k \in L - H(V_0)$ for every $k > m(n)$. Prove that $U_{n+1} \cap V_0 \neq \varnothing$, and conclude that $X$ is not normal.

14) Deduce from Exercise 13 that: (a) If a product $\prod_{i \in I} X_i$ of separated spaces is normal, $X_i$ is semi-compact (IX, p. 93, Exercise 13) unless for a countable set of indices. (b) For a product $\prod_{i \in I} X_i$ of metrizable spaces to be normal it is necessary and sufficient that $X_i$ be compact unless for a countable set of indices; the product space is then paracompact.