Exercise 20, § 4

For every covering $\mathscr{R}$ of a topological space $X$, we denote by $V_{\mathscr{R}}$ the union of the sets $U \times U$, where $U$ runs through $\mathscr{R}$. A covering $\mathscr{R}$ of $X$ is said to be uniform if there exists a neighbourhood $W$ of the diagonal $\Delta$ of $X \times X$ such that the covering formed by the $W(x)$, where $x$ runs through $X$, is finer than $\mathscr{R}$. A covering $\mathscr{R}$ is said to be divisible if there exists a neighbourhood $W$ of $\Delta$ in $X \times X$ such that $W \subset V_{\mathscr{R}}$.

a) Prove that every uniform open covering of $X$ is divisible (cf. IX, p. 107, exerc. 26 b)).

b) Let $\mathscr{R}$ be an open covering of $X$ such that there exists a locally finite closed covering $\mathscr{S}$ of $X$, finer than $\mathscr{R}$. Prove that the covering $\mathscr{R}$ is uniform. (For every set $A \in \mathscr{S}$, let $U_A$ be a set of $\mathscr{R}$ containing $A$ and let $V_A$ be the union in $X \times X$ of the open sets $U_A \times U_A$ and $(X - A) \times (X - A)$; consider the set $W = \bigcap_{A \in \mathscr{S}} V_A$).

c) Let $X$ be a completely regular space; a covering $\mathscr{R}$ of $X$ is said to be normal if $V_{\mathscr{R}}$ is an entourage of the universal uniform structure $\mathscr{U}_\omega$ (IX, p. 86, exerc. 10) of $X$; it is equivalent to saying that there exists a continuous distance $d$ on $X \times X$, such that the set of the $(x, y) \in X \times X$ such that $d(x, y) < 1$ is contained in $V_{\mathscr{R}}$. There then exists an infinite sequence $(\mathscr{R}_n)$ of coverings of $X$ such that $\mathscr{R}_1 = \mathscr{R}$ and $V_{\mathscr{R}_{n+1}}^2 \subset V_{\mathscr{R}_n}$ for every $n$; in particular, every normal open covering is uniform.

d) Let $X$ be a completely regular space, and let $(U, V)$ be an open covering of $X$ (such a covering is said to be binary). In order that $(U, V)$ be a divisible covering, it is necessary and sufficient that, if one sets $A = X - U, B = X - V$, there exist two disjoint open sets $S, T$ such that $A \subset S$ and $B \subset T$. In order that $(U, V)$ be a normal covering, it is necessary and sufficient that $A$ and $B$ be normally separated (IX, p. 103, exerc. 15). Deduce that for every $x \in X$ and every open neighbourhood $U$ of $x$, there exists an open neighbourhood $V$ of $x$ such that $\overline{V} \subset U$ and that $(U, X - \overline{V})$ be a normal covering of $X$.

e) In order that a completely regular space $X$ be Weierstrassian (IX, p. 88, exerc. 20), it is necessary and sufficient that every normal open covering of $X$ contain a finite covering (use IX, p. 89, exerc. 20 d)).