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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

### Keywords:

continuous of first-class selections; \(F_{\sigma }\) set; derived set of order \(\alpha \); length; dispersed space; hereditarily strongly paracompact; zero-dimensional; Vietoris topology; continuous selection
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\textit{O. N. Kolesnikov}, Sib. Math. J. 27, 55--62 (1986; Zbl 0625.54022); translation from Sib. Mat. Zh. 27, No. 1, 70--78 (1986)

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### References:

[1] | E. Michael, ?Selected selection theorem,? Am. Math. Mon.,63, 361 (1956). · Zbl 0070.39502 |

[2] | M. M. Choban, ?Multivalued maps and Borel sets. I, II,? Tr. Mosk. Mat. Obshch.,22, 229-250 (1970);23, 277-301 (1970). · Zbl 0231.54013 |

[3] | J. van Mill and E. Wattel, ?Selections and orderability,? Proc. Am. Math. Soc.,83, No. 3, 601-605 (1981). · Zbl 0473.54010 |

[4] | N. K. Dodon, ?Subsets with the property?, and the topology of?-rarefied spaces,? Vestn. Mosk. Gos. Univ., Ser. Mat., No. 6, 11-18 (1977). |

[5] | M. M. Choban and N. K. Dodon, The Theory of P-Rarefied Spaces [in Russian], Kishinev (1979). · Zbl 0506.54029 |

[6] | B. A. Pasynkov, ?A factorization theorem for nonclosed sets,? Dokl. Akad. Nauk SSR,202, No. 6, 1274-1276 (1972). |

[7] | G. D. Creede, ?Concerning semistratifiable spaces,? Pac. J. Math.,32, No. 1 (1970). |

[8] | H. J. Junnila, ?Stratifiable preimages of topological spaces,? Colloq. Math. Soc. J. Bolyai,23, Topology, Budapest (1978), pp. 689-703. |

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