Exercise 19, § 4

a) A regular space $X$ is paracompact if and only if every open covering of $X$ is even (Exercise 16). [To prove necessity, use Lemma 5 and Exercise 16 b); for sufficiency, use Exercise 16 a) of $§ 4$, Proposition 2 of $§ 1$, no. 4, and Theorem 4 of $§ 4$, no. 5].

b) Give an example of a divisible non-even covering of a collectively normal, non-paracompact space [use a) and Chapter II, § 4, Exercise 4].

c) Show that a paracompact space $X$ is complete with respect to its universal uniformity ($§ 1$, Exercise 5). [If a Cauchy filter $\mathfrak{F}$ with respect to this uniformity has no cluster point, then every point $x \in X$ has an open neighbourhood $V_x$ which does not meet at least one set of $\mathfrak{F}$; use a) and Exercise 18 a).]