Exercise 16, § 4

Let $(X_i)_{i \in I}$ be a non-denumerable family of separated topological spaces, each having at least two distinct points. For every $i \in I$, let $a_i, b_i$ 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 $X$ formed by the points $(x_i)_{i \in I}$ such that $x_i = a_i$ except for a denumerable set of indices $i$; let $b$ be the point $(b_i)$ of $X$. In the product space $X \times Y$, consider the sets $A = Y \times Y$ and $B = \{b\} \times Y$. Now prove that $A$ and $B$ are closed and that there is no pair of open sets $U, V$ in $X \times Y$ such that $A \subset U, B \subset V$ and $U \cap V = \varnothing$. (Let $U, V$ be two open sets such that $A \subset U$ and $B \subset V$; show that there exists an increasing sequence $(H_n)$ of denumerable subsets of $I$, such that, if one denotes by $x_n$ the point of $Y$ whose every coordinate of index $i \in H_n$ is equal to $b_i$, every coordinate of index $i \notin H_n$ equal to $a_i$, the point $(x_{n+1}, x_n)$ belongs to $V$ for every integer $n$; proving that the sequence $(x_{n+1}, x_n)$ tends to a point of $A$, and conclude that $U \cap V \neq \varnothing$). Deduce from this result an example of a connected and non-normal topological ring. Also deduce that the product of a non-denumerable family of separated topological spaces each of which has at least two distinct points cannot be completely normal (use the result above when all the $X_i$ are identical to a discrete space formed of two points.)