A countably compact, separable space which is not absolutely countably compact. (English) Zbl 0833.54012

Vaughan, Jerry E.

Commentat. Math. Univ. Carol. 36, No. 1, 197-201 (1995).

A space \(X\) is absolutely countably compact provided for every open cover \({\mathcal U}\) of \(X\) and for every dense subspace \(Y\subset X\) there exists a finite subset \(A\subset Y\) such that \(St (A,{\mathcal U})= X\) (if we remove “for every dense subspace \(Y\subset X\)” and write \(A\subset X\) instead of \(A\subset Y\), then we obtain a weaker condition, starcompactness, which is known to be equivalent to countable compactness in the class of Hausdorff spaces). Answering a question of the reviewer, the author constructs a space having the properties in the title. Also, he gives an example of a countably compact topological group which is not absolutely countably compact. Both examples are derived from the following interesting observation: if a \(T_1\) space \(X\) has an open cover \({\mathcal U}\) which does not have a finite subcover, then the product space \(X^{\mathfrak k}\), where \({\mathfrak k}= |{\mathcal U}|\), is not absolutely countably compact.

Reviewer: M.V.Matveev (Moskva)

MSC:

54D20	Noncompact covering properties (paracompact, Lindelöf, etc.)
54B10	Product spaces in general topology
54G20	Counterexamples in general topology
54H11	Topological groups (topological aspects)

Keywords:

absolutely countably compact space; starcompactness; countable compactness; countably compact topological group