A countably compact, separable space which is not absolutely countably compact.

*(English)*Zbl 0833.54012A 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, LindelĂ¶f, etc.) |

54B10 | Product spaces in general topology |

54G20 | Counterexamples in general topology |

54H11 | Topological groups (topological aspects) |