On relatively almost countably compact subsets. (English) Zbl 1224.54054

Song, Yan-Kui; Zheng, Shu-Nian

Math. Bohem. 135, No. 3, 291-297 (2010).

Summary: A subset \(Y\) of a space \(X\) is almost countably compact in \(X\) if, for every countable cover \(\mathcal U\) of \(Y\) by open subsets of \(X\), there exists a finite subfamily \(\mathcal V\) of \(\mathcal U\) such that \(Y\subseteq \overline {\bigcup \mathcal V}\). In this paper, we investigate the relationship between almost countably compact spaces and relatively almost countably compact subsets, and we also study various properties of relatively almost countably compact subsets.

MSC:

54D15	Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)
54D20	Noncompact covering properties (paracompact, Lindelöf, etc.)

Keywords:

countably compact space; almost countably compact space; relatively almost countably compact subset