Exercise 36, § 4

It is said that a topological space X is locally paracompact if every point x ∈ X admits in X a closed neighbourhood which is a paracompact subspace.

a) Now prove that a locally paracompact space is completely regular.

b) In the Tychonoff plank P (IX, p. 85, exerc. 8), let H be the complement of the set of points (n, ω₁). Now prove that H is a normal space which is not locally paracompact (cf. I, p. 106, exerc. 12).

c) Let X be a topological space, Φ a covering of X formed of closed sets which are paracompact subspaces of X and are such that every A ∈ Φ admits a neighbourhood in X belonging to Φ. Let X’ be the sum set of X and of a point ω; one defines on X’ a topology by taking as fundamental system of neighbourhoods of x ∈ X in X’ the set of neighbourhoods of x in X, and as fundamental system of neighbourhoods of ω the filter basis generated by the complements in X’ of the sets of Φ (cf. IX, p. 109, exerc. 30)). Now prove that X’ is a paracompact space.