Exercise 25, § 4

A Hausdorff space $X$ is said to be metacompact if, given any open covering $R$ of $X$, there is a point-finite open covering of $X$ which refines $R$.

a) Show that every closed subspace of a metacompact space is metacompact, and that if every open subspace of a metacompact space $X$ is metacompact, then every subspace of $X$ is metacompact.

b) Let $X$ be a Hausdorff space and let $R$ be a closed equivalence relation on $X$, all of whose equivalence classes are compact. Show that if $X/R$ is metacompact, then $X$ is metacompact [argue as in Exercise 21 b)].

c) Show that a topological space which is both metacompact and countably compact (§ 2, Exercise 14) is compact. [Use Zorn’s lemma to show that every point-finite open covering contains a minimal open covering, and use § 2, Exercise 14 a).]

d) The locally compact, completely normal space $X = [a, b[$ defined in Chapter I, § 9, Exercise 12 is not metacompact (*).

e) Let $Y$ be a metrizable space and let $A$ be a subset of $Y$ such that both $A$ and its complement are dense in $Y$. Let $X$ denote the non-regular space obtained by endowing $Y$ with the topology generated by the open sets of $Y$ and the set $A$. Show that $X$ is metacompact. (For each $x \in X$, consider a neighbourhood $V_x$ of $x$ contained in a set of a given open covering $\mathfrak{R}$ of $X$, such that $V_x \subset A$ if $x \in A$, and such that $V_x$ is a neighbourhood of $x$ in $Y$ if $x \notin A$. Note that the subspace $A$ of $X$ is paracompact, and that the union of the $V_x$ for $x \notin A$ is also paracompact.)