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Selections of multivalued maps with values in rarefied spaces. (English. Russian original) Zbl 0625.54022

Sib. Math. J. 27, 55-62 (1986); translation from Sib. Mat. Zh. 27, No. 1, 70-78 (1986).
The author considers continuous and semicontinuous multivalued maps and finds conditions under which they have continuous of first-class selections. A map \(f: X\to Y\) is first-class if the preimage of every open set Y is an \(F_{\sigma}\) set in X. He defines the derived set of order \(\alpha\) of the space Y by the conditions: \(Y^{(1)}=Y^ d\) (where \(Y^ d\) is the usual derived set), \(Y^{(\alpha +1)}=(Y^{(\alpha)})^ d\), and if \(\lambda\) is a limit ordinal \(Y^{(\lambda)}=\cap_{\alpha <\lambda}Y^{(\alpha)}\). A space Y is dispersed if there is an \(\alpha\) such that \(Y^{(\alpha)}=\emptyset\), and the smallest such \(\alpha\) is the length of the space. The proofs of all of the fifteen theorems are carried out by induction on the length of the dispersed space Y. Sample theorem: If X is hereditarily strongly paracompact, Y is inductively zero-dimensional and dispersed, and \(F: X\to A(Y)\) (A(Y) is the set of all nonempty subsets of Y with the Vietoris topology), then F has a continuous selection.
Reviewer: R.P.Jerrard

MSC:

54C65 Selections in general topology
54C60 Set-valued maps in general topology
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References:

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