## 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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