Exercise 23, § 4

(a) In order that every finite open covering of a completely regular space be divisible (IX, p. 105, exerc. 20), it is necessary that X be normal; conversely, if X is normal, every finite open covering of X is normal (IX, p. 105, exerc. 20); when $\mathscr{R}$ runs through the set of finite open coverings of X, the sets $V_{\mathscr{R}}$ form a fundamental system of entourages of the uniform structure $\mathscr{U}_\omega^*$ induced on X by the uniform structure of $\beta X$ (Use IX, p. 48, cor. 1 and axiom (Ov) to define a finite family of continuous distances on X of the form $(x, y) \mapsto |f(x) - f(y)|$, in such a way that an entourage of the uniform structure defined by this family is contained in $V_{\mathscr{R}}$). (b) Let X be a normal space; for every closed subset Y of X, let $Y^\beta$ be the closure of Y in $\beta X$. Now prove that the continuous mapping $\beta Y \to Y^\beta$ which extends the canonical injection $Y \to Y^\beta$ is a homeomorphism (use a), or directly the universal property of $\beta Y$ and th. 2 of IX, p. 44). If Y, Z are two closed subsets of X, prove that

in $\beta X$ (if $x_0 \notin (Y \cap Z)^\beta$ and $x_0 \in Y^\beta$, consider a closed neighbourhood V of $x_0$ in $\beta X$, such that $V \cap Y$ and Z are disjoint).