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The shrinking property. (English) Zbl 0536.54013

Author’s abstract. ”A space has the shrinking property if, for every open cover \(\{V_ a| a\in A\}\), there is an open cover \(\{W_ a| a\in A\}\) with \(W_ a\subset V_ a\) for each \(a\in A\). It is strangely difficult to find an example of a normal space without the shrinking property. It is proved here that any \(\Sigma\)-product of metric spaces has the shrinking property.” In addition to an intricate proof of the above theorem, the author elaborates on the second sentence noting that Dowker spaces do not have shrinking covers. Also, reference is given to some related results coming out by A. Le Donne, Normality and shrinking property in \(\Sigma\)-product of spaces, Can. Math. Bull. (to appear).
Reviewer: C.E.Aull

MSC:

54D20 Noncompact covering properties (paracompact, Lindelöf, etc.)
54B10 Product spaces in general topology
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