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**Sur les selecteurs des multifonctions. (On selections of multifunctions).**
*(French)*
Zbl 0629.54013

The following question appears in specific forms in various fields of mathematics: How “continuous” resp. “measurable” a multifunction F between two sufficiently structured spaces X and Y has to be in order to have a sequence of “continuous” resp. “measurable” selectors \(\{f_ n\}_{n\in {\mathbb{N}}}\) such that \(F(x)=cl\cup^{\infty}_{n=I}f_ n(x)\) for all x in X? The full relevance of this question can be seen in the light of Kuratowski’s “Topology”, of C. Castaing’s 1967 “Thesis” [see also Rev. Franç. Inform. Rech. Opér. 1, 91-126 (1967; Zbl 0153.085)] and of their developments. The reviewed paper gives positive answers to this question for regular topological spaces X and Y while F is either upper semicontinuous and 1-quasi-continuous or it reverses open subsets of Y to Baire subsets of X (Baire measurability). In context, important properties of Baire measurable functions are revealed.

Reviewer: D.Butnariu

### MSC:

54C60 | Set-valued maps in general topology |

28B20 | Set-valued set functions and measures; integration of set-valued functions; measurable selections |

54C05 | Continuous maps |

54C65 | Selections in general topology |

54C50 | Topology of special sets defined by functions |

### Keywords:

Baire measurable functions### Citations:

Zbl 0153.085
Full Text:
EuDML

### References:

[1] | NEUBRUNNOVÁ A.: On quasicontinuous and cliquish functions. Čas. pěst. mat. 99, 1974, 109-114. · Zbl 0292.26005 |

[2] | NEUBRUNN T., NÁTHER O.: On a characterization of quasicontinuous multifunctions. Čas. pěst. mat. 107, 1982, 294-300. · Zbl 0532.54016 |

[3] | KURATOWSKI C.: Topologoe I. Warszawa 1952. |

[4] | KURATOWSKI C.: Topologie II. Warszawa 1952. · Zbl 0049.39704 |

[5] | KURATOWSKI K., MOSTOWSKI A.: Teoria mnogosci. Warszawa 1978. |

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