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Solution to a compactification problem of Sklyarenko. (English) Zbl 0671.54014

The compactness deficiency of a (metric separable) space is the least integer n for which the space has a compactification whose remainder is of dimension n. Another measure of the defect of compactness is the number introduced by E. G. Sklyarenko [Tr. Tbilis. Mat. Inst. 27, 113-114 (1960; Zbl 0151.301) and a paper in Transl. Am. Math. Soc., II. Ser. 58, 216-244 (1966; Zbl 0178.566)], namely the minimal number among those for which the space has a base such that the intersection of boundaries of arbitrary \(n+1\) members of the base is compact. Sklyarenko proved that this last number does not exceed the compactness defiency. The author proves the reverse inequality, thus both these measures of the defect of compactness are equal.
Reviewer: J.Mioduszewski

MSC:

54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54E45 Compact (locally compact) metric spaces
54F45 Dimension theory in general topology
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