Exercise 17, § 4

a) If every finite open covering of a topological space $X$ is divisible (Exercise 16), then $X$ satisfies Axiom $(O'_V)$ (consider a binary open covering of $X$, i.e., one formed by two subsets of $X$). Conversely, if $X$ satisfies $(O'_V)$, then every finite open covering of $X$ is even [use Theorem 3 of no. 3, which is valid provided that $X$ satisfies $(O'_V)$, to reduce to the case of a binary covering, and then again to deal with this case]. As $\mathfrak{R}$ runs through the set of finite open coverings of $X$, the sets $V_{\mathfrak{R}}$ form a fundamental system of entourages of a uniformity on $X$, compatible with the topology of $X$; this uniformity is called the "uniformity of finite open coverings".

b) Suppose that $X$ is normal. Show that the uniformity $\mathscr{U}$ of finite open coverings is the same as the coarsest uniformity $\mathscr{U}'$ for which all continuous mappings of $X$ into $[0, 1]$ are uniformly continuous [i.e., the uniformity induced on $X$ by that of its Stone-\v{C}ech compactification ($§ 1$, Exercise 7)]. [To show that $\mathscr{U}$ is coarser than $\mathscr{U}'$, use Theorem 3 of no. 3 and Axiom $(O_V)$ in order to construct a finite family of pseudometrics on $X$ of the form $(x, y) \to |f(x) - f(y)|$, in such a way that an entourage of the uniformity defined by these pseudometrics is contained in an entourage of the uniformity $\mathscr{U}$.

c) Let $X$ be a normal space and let $\beta X$ be its Stone-\v{C}ech compactification. For each closed subset $Y$ of $X$, let $\overline{Y}$ be the closure of $Y$ in $\beta X$. Show that the continuous mapping of $\beta Y$ into $\overline{Y}$ which extends the identity mapping of $Y$ ($\beta Y$ being the Stone-\v{C}ech compactification of $Y$) is a homeomorphism [either use b) or else Theorem 2 and the universal property of $\beta Y$]. If $Y$ and $Z$ are two closed subsets of $X$, show that $\overline{Y \cap Z} = \overline{Y} \cap \overline{Z}$ in $\beta X$ (if $x_0 \notin \overline{Y \cap Z}$ and $x_0 \in \overline{Y}$, consider a closed neighbourhood $V$ of $x_0$ in $\beta X$, such that $V \cap Y$ and $Z$ are disjoint closed subsets of $X$).