Exercise 13, § 4

Let $(X_i)_{i \in I}$ be an uncountable family of Hausdorff topological spaces, each containing more than one point. For each $i \in I$, let $a_i$ and $b_i$ be two distinct points of $X_i$. Let $X$ be the product space $\prod_{i \in I} X_i$, and let $Y$ be the subspace of $X$ consisting of these points $(x_i)_{i \in I}$ such that $x_i = a_i$ for all but a countable set of indices $i$; and let $b$ be the point $b_i$ of $X$.

In the product space $X \times Y$, let $A$ be the diagonal of $Y \times Y$ and let $B$ be the set $\{b\} \times Y$. Show that $A$ and $B$ are closed sets and that there is no pair of open sets $U, V$ in $X \times Y$ such that $A \subset U, B \subset V$, and $U \cap V = \emptyset$. [Let $U, V$ be two open sets such that $A \subset U$ and $B \subset V$; show that there is an increasing sequence $(H_n)$ of countable subsets of $I$, such that if $x_n$ denotes the point of $Y$ for which each coordinate with index $i \in H_n$ is equal to $b_i$, and each coordinate with index $i \notin H_n$ is equal to $a_i$, then the point $(x_{n+1}, x_n)$ belongs to $V$ for each integer $n$; prove that the sequence $(x_{n+1}, x_n)$ tends to a point of $A$, and deduce that $U \cap V \neq \emptyset$.]

Hence construct an example of a connected topological ring which is not normal.

Deduce also that the product of an uncountable family of Hausdorff topological spaces, each of which has at least two distinct points, cannot be completely normal (use the result above when each $X_i$ is a discrete space of two points).