Exercise 16, § 4

For each covering $\mathfrak{R}$ of a topological space $X$, let $V_{\mathfrak{R}}$ denote the union of all the sets $U \times U$ such that $U \in \mathfrak{R}$. A covering $\mathfrak{R}$ of $X$ is said to be even if there is a neighbourhood $W$ of the diagonal $\Delta$ of $X \times X$ such that the covering $(W(x))_{x \in X}$ is finer than $\mathfrak{R}$. $\mathfrak{R}$ is said to be divisible if there is a neighbourhood $W$ of $\Delta$ in $X \times X$ such that $\dot{W} \subset V_{\mathfrak{R}}$.

a) Show that every even open covering of $X$ is divisible [cf. Exercise 19 b)].

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