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Čech-completeness and ultracompleteness in “nice” spaces. (English) Zbl 1090.54023
A property \({\mathcal P}\) is called (i) additive if \(X\) enjoys \({\mathcal P}\) whenever it can be covered by a finite number of subspaces with \({\mathcal P}\); (ii) \(n\)-additive in power \(n\) if \(X\) enjoys \({\mathcal P}\) whenever \(X^n\) can be covered by \(n\) subsets with property \({\mathcal P}\); (iii) additive in finite powers if it is \(n\)-additive in power \(n\) for each \(n\).
The authors study these kinds of additivity for various classes of complete spaces. They prove, for example, that the property of pointwise countable type is additive in finite powers, as well as ultracompleteness within metrizable spaces. On the other hand, ultracompleteness is not additive even within separable metrizable spaces while any space covered by finitely many dense ultracomplete subspaces is ultracomplete. Another result is a construction (under CH) of a subspace of \(\beta \,\omega \) which is countably compact and Čech-complete but not ultracomplete. The paper is finished by eleven open problems.
MSC:
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54D70 Base properties of topological spaces
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