On a Core Concept of Arhangel’skĭı

Arhangel’skĭı [3] has introduced a weakening of σ-compactness: having a countable core, for locally compact spaces, and asked when it is equivalent to σ-compactness. We settle several problems related to that paper. The concept of countable core in [3] is a little hard to understand at first; Arhangel’skĭı, however, provides equivalents which are easier to understand, and so we will take one of them as our definition, referring the reader to [3] for the original definition. Definition. A subset Y of a space X is compact from inside if every subspace F of Y which is closed in X is compact. A locally compact space X has a countable core if it has a countable open cover by sets compact from inside X. The motivation for considering this concept lies in considering the implications of the point at infinity in the one-point compactification of a locally compact space having various local countability properties — see the following Definition, Proposition, and Lemma. Let a be the point at infinity in the one-point compactification aX of a locally compact space X (we shall assume all spaces are Hausdorff). Research supported by NSERC grant A-7354. 2000 Math. Subj. Class. Primary 54D45, 54D20, 54A35, 54A25; Secondary 54G20, 54D65, 54D55, 03E35, 03E50, 03E65.