Exercise 27, § 4

A topological space $X$ is said to be locally paracompact if every point of $X$ has a closed neighbourhood which is a paracompact subspace of $X$.

(*) There are spaces which are perfectly normal and collectively normal but not metacompact, and spaces which are perfectly normal and metacompact but not collectively normal (and therefore not paracompact); cf. E. Michael, Can. Journ. of Math. 7 (1955), pp. 275-279.

a) Show that a locally paracompact space is completely regular.

b) In the compact space $Y_0 \times Z$ defined in Exercise 26 b), let $H$ be the complement of the set of points $(b, y)$ where $y < c$. Show that $H$ is normal but not locally paracompact [cf. Chapter I, § 9, Exercise 12 b)].

c) Let $X$ be a topological space and let $\Phi$ be a covering of $X$ by closed sets which are paracompact subspaces of $X$, such that each $A \in \Phi$ has a neighbourhood in $X$ which belongs to $\Phi$. Let $X'$ be the set which is the sum of $X$ and a point $\omega$; define a topology on $X'$ by taking as a fundamental system of neighbourhoods of $x \in X$ in $X'$ the set of all neighbourhoods of $x$ in $X$, and as a fundamental system of neighbourhoods of $\omega$ the filter base generated by the complements in $X'$ of the sets of $\Phi$ [cf. Exercise 20 c)]. Show that $X'$ is paracompact.