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Sums and products of ultracomplete topological spaces. (English) Zbl 1019.54015
In a previous paper [Acta Math. Hung. 92, 19-26 (2001; Zbl 0997.54037)] the authors showed that strongly complete topological spaces (introduced by V. I. Ponomarev and V. V. Tkachuk) and cofinally Čech complete topological spaces (introduced by S. Romaguera) are equivalent notions and called such spaces ultracomplete. In the present paper the authors study sums and products of such spaces. Sample result: Theorem 4.12. Let $$X$$ and $$Y$$ be two ultracomplete spaces. Then $$X\times Y$$ is ultracomplete if and only if one of the following conditions holds: (i) both $$X$$ and $$Y$$ are locally compact, or (ii) either $$X$$ or $$Y$$ is countably compact, locally compact, or (iii) both $$X$$ and $$Y$$ are countably compact. Further, the authors ask whether each Čech complete, countably compact space is ultracomplete. They show that for $$GO$$ spaces the answer is yes.

##### MSC:
 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.) 54B10 Product spaces in general topology 54D99 Fairly general properties of topological spaces
Zbl 0997.54037
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