Exercise 26, § 4

a) For a regular space X to be paracompact, it is necessary and sufficient that every open covering of X be uniform (IX, p. 105, Exercise 20). (To see that the condition is necessary, use Exercise 25 a) and IX, p. 105, Exercise 20 b). To see that it is sufficient, use IX, p. 105, Exercise 20 a), IX, p. 6, Proposition 2 and IX, p. 51, Theorem 4.) Every paracompact space is therefore collectively normal (IX, p. 106, Exercise 24) and the set of neighborhoods of the diagonal of $X \times X$ is the filter of entourages of the universal uniform structure on X. b) Give an example of a divisible non-uniform covering of a locally compact, collectively normal and non-paracompact space (use a), and II, p. 37, Exercise 4). c) Show that a paracompact space X is complete for its universal uniform structure (IX, p. 86, Exercise 10). (If a Cauchy filter $\mathscr{F}$ for this structure has no adherent point, every point $x \in X$ has an open neighborhood $V_x$ which does not meet at least one set of $\mathscr{F}$. Use then a), and IX, p. 106, Exercise 24 a).)