Pseudocompact metacompact spaces are compact

Abstract

In 1950, Arens and Dugundji [1] defined metacompact spaces and showed (A) countably compact metacompact spaces are compact. It was known by then that paracompact spaces are normal and that normal pseudocompact spaces are countably compact. Since paracompact spaces are metacompact, (A) also showed (B) pseudocompact paracompact spaces are compact. The results (A) and (B) raise the following question of Aull and others [2]. as a common generalization: are pseudocompact metacompact spaces compact? This question was answered in the affirmative, first by Scott [3] and independently by Forster [4] and the author. The short proof given here also establishes an interesting property of Baire spaces. All spaces are assumed completely regular. A space is pseudocompact if every continuous real-valued function on it is bounded. A space is Baire if no open set is the union of countably many nowhere dense sets. A 7r-base for a space X is a family S of nonempty open subsets of X such that if G is a nonempty open subset of X, then some element of 3 is contained in G.