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Near metric properties of hyperspaces. (English) Zbl 0923.54013

By a well-known theorem of N. V. Velichko [Sib. Mat. Zh. 16, 627-629 (1975; Zbl 0317.54007)] the normality of the hyperspace \(2^X\) of a space \(X\) is equivalent to the compactness of \(X\); probably less well-known is that hereditary normality of \(2^X\) is equivalent to compactness plus metrizability of \(X\). These results show that ‘near metric’ properties of \(2^X\) will be equivalent to compact metrizability of \(X\) (and \(2^X\)). Accordingly the authors of the present paper concentrate on two subspaces of \(2^X\): the space of compact subsets \({\mathcal K}(X)\) and the space of finite sets \({\mathcal F}(X)\). Sample results are: \({\mathcal F}(X)\) is monotonically normal iff all of its finite powers are, iff all finite powers of \(X\) are, and if every open set of \(X\) contains an infinite compact subset then \({\mathcal K}(X)\) is monotonically normal iff it is stratifiable.
Reviewer: K.P.Hart (Delft)

MSC:

54B20 Hyperspaces in general topology
54E20 Stratifiable spaces, cosmic spaces, etc.
54D15 Higher separation axioms (completely regular, normal, perfectly or collectionwise normal, etc.)

Citations:

Zbl 0317.54007
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