Exercise 35, § 4

One says that a separated space X is metacompact if for every open covering $\mathscr{R}$ of X, there exists an open covering of X finer than $\mathscr{R}$ and point-finite.

(a) Now prove that every closed subspace of a metacompact space is metacompact; if every open subspace of a metacompact space X is metacompact, then every subspace of X is metacompact.

(b) Let X be a separated space, R a closed equivalence relation in X such that every equivalence class according to R is compact. Now prove that if X/R is metacompact, X is metacompact (reason as in IX, p. 109, exerc. 31 b)).

(c) Now prove that a space which is at the same time metacompact and semi-compact (IX, p. 93, exerc. 13) is compact. (Remark, with the aid of Zorn's theorem, that every point-finite open covering contains a minimal open covering and use exerc. 13 a) of IX, p. 93).

(d) The locally compact and completely normal space $X = (a, b[$ defined in exerc. 12 of I, p. 106, is not metacompact¹.

(e) Let Y be a metrizable space, A a subset of Y dense everywhere as is its complement in Y. We denote by X the non-regular space obtained by endowing Y with the topology generated by the open sets of Y and the set A (I, p. 103, exerc. 20). Now prove that X is metacompact. (Consider for every $x \in X$ a neighbourhood $V_x$ of x contained in an element of a given open covering $\mathscr{R}$ of X, such that $V_x \subset A$ if $x \in A$ and that $V_x$ is a neighbourhood of x in Y if $x \notin A$. Remark that A, considered as a subspace of Y, is paracompact, and that the same is true of the union of the $V_x$ for $x \in A$, considered as a subspace of Y.)

¹ There are spaces which are perfectly normal, collectively normal and non-metacompact, and spaces which are perfectly normal, metacompact and non-collectively normal (therefore non-paracompact); cf. E. Michael, Can. Journ. of Math., t. VII (1955), p. 275–279.