Chapter 25 - First countable, countably compact, noncompact spaces

Abstract

This chapter focuses on a problem related to first countable, countably compact, noncompact spaces. This problem asks—“Does ZFC imply the existence of a separable, first countable, countably compact, noncompact Hausdorff (T2) space?” The usual topology on ω1 satisfies everything except separability. The Novak–Teresaka space described in Vaughan's article satisfies everything except first countability. If the co-finite topology on ω1 is refined by making initial segments open, then the resulting space satisfies everything except T2 and is T1. The remaining two properties are obviously necessary also to have an open problem. The problem mentioned is one of a small but growing number of topological problems for which a negative answer is known to entail (2ω =)c ≥א3, yet c =א3 has not been ruled out. The chapter discusses that a space X is ω-bounded if every countable subset has compact closure, and strongly ω-bounded if every σ-compact subset has compact closure. The chapter describes basic concepts related to good spaces. It also provides details about other consistent constructions for the stated problem.