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

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