Exercise 10, § 4

Let $L$ be an uncountable set and let $X$ denote the completely regular space $N^L$ ($N$ carrying the discrete topology). Let $A$ (resp. $B$) be the set of all $x = (x(\lambda))_{\lambda \in L}$ in $X$ such that, for each integer $k \neq 0$ (resp. $k \neq 1$) the set of $\lambda \in L$ such that $x(\lambda) = k$ has at most one element.

a) Show that $A$ and $B$ are disjoint closed subsets of $X$.

b) Let $U, V$ be two open subsets of $X$ such that $A \subset U$ and $B \subset V$. Let $\Phi$ denote the set of all elementary sets in $X$ (Chapter I, § 4, no. 1) whose projections which are distinct from $N$ consist of a single element; for each $W \in \Phi$, let $H(W)$ be the set of all $\lambda \in L$ such that $pr_\lambda(W)$ consists of a single element. Show that there is 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 with the following properties: (i) $U_n \subset U$ is a neighbourhood of $x_n$; (ii) $H(U_n)$ is the set of all $\lambda_k$ such that $k \leq m(n)$; (iii) $x_0(\lambda) = 0$ for each $\lambda \in L$, and for each $n > 0$, $x_n(\lambda_k) = k$ if $k \leq m(n-1)$ and $x_n(\lambda) = 0$ for all other $\lambda \in L$.

c) Let $y \in B$ be the point such that $y(\lambda_k) = k$ for each integer $k$ and $y(\lambda) = 1$ for all $\lambda \in L$ other than the $\lambda_k$. Let $V_0 \subset V$ be a set of $\Phi$ which contains $y$, and let $n$ be an integer such that $\lambda_k \in L - H(V_0)$ for all $k > m(n)$. Show that $U_{n+1} \cap V_0 \neq \varnothing$, and deduce that $X$ is not normal.