# Selections and representations of multifunctions in paracompact spaces

Studia Mathematica (1992)

• Volume: 102, Issue: 3, page 209-216
• ISSN: 0039-3223

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## Abstract

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Let (X,T) be a paracompact space, Y a complete metric space, $F:X\to {2}^{Y}$ a lower semicontinuous multifunction with nonempty closed values. We prove that if ${T}^{+}$ is a (stronger than T) topology on X satisfying a compatibility property, then F admits a ${T}^{+}$-continuous selection. If Y is separable, then there exists a sequence $\left({f}_{n}\right)$ of ${T}^{+}$-continuous selections such that $F\left(x\right)=\overline{{f}_{n}\left(x\right);n\ge 1}$ for all x ∈ X. Given a Banach space E, the above result is then used to construct directionally continuous selections on arbitrary subsets of ℝ × E.

## How to cite

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Bressan, Alberto, and Colombo, Giovanni. "Selections and representations of multifunctions in paracompact spaces." Studia Mathematica 102.3 (1992): 209-216. <http://eudml.org/doc/215923>.

@article{Bressan1992,
abstract = {Let (X,T) be a paracompact space, Y a complete metric space, $F:X → 2^Y$ a lower semicontinuous multifunction with nonempty closed values. We prove that if $T^+$ is a (stronger than T) topology on X satisfying a compatibility property, then F admits a $T^+$-continuous selection. If Y is separable, then there exists a sequence $(f_n)$ of $T^+$-continuous selections such that $F(x)=\overline\{\{f_n(x);n ≥ 1\}\}$ for all x ∈ X. Given a Banach space E, the above result is then used to construct directionally continuous selections on arbitrary subsets of ℝ × E.},
author = {Bressan, Alberto, Colombo, Giovanni},
journal = {Studia Mathematica},
keywords = {directionally continuous selections},
language = {eng},
number = {3},
pages = {209-216},
title = {Selections and representations of multifunctions in paracompact spaces},
url = {http://eudml.org/doc/215923},
volume = {102},
year = {1992},
}

TY - JOUR
AU - Bressan, Alberto
AU - Colombo, Giovanni
TI - Selections and representations of multifunctions in paracompact spaces
JO - Studia Mathematica
PY - 1992
VL - 102
IS - 3
SP - 209
EP - 216
AB - Let (X,T) be a paracompact space, Y a complete metric space, $F:X → 2^Y$ a lower semicontinuous multifunction with nonempty closed values. We prove that if $T^+$ is a (stronger than T) topology on X satisfying a compatibility property, then F admits a $T^+$-continuous selection. If Y is separable, then there exists a sequence $(f_n)$ of $T^+$-continuous selections such that $F(x)=\overline{{f_n(x);n ≥ 1}}$ for all x ∈ X. Given a Banach space E, the above result is then used to construct directionally continuous selections on arbitrary subsets of ℝ × E.
LA - eng
KW - directionally continuous selections
UR - http://eudml.org/doc/215923
ER -

## References

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1. [1] J. P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin 1984.
2. [2] A. Bressan,HAMUpper and lower semicontinuous differential inclusions. A unified approach, in: Controllability and Optimal Control, H. Sussmann (ed.), M. Dekker, New York 1989, 21-32.
3. [5] A. Bressan and G. Colombo, Boundary value problems for lower semicontinuous differential inclusions, Funkcial. Ekvac., to appear. Zbl0788.34007
4. [6] A. Bressan and A. Cortesi, Directionally continuous selections in Banach spaces, Nonlin. Anal. 13 (1989), 987-992. Zbl0687.34013
5. [7] E. Michael, Selected selection theorems, Amer. Math. Monthly 63 (1956), 233-238. Zbl0070.39502
6. [8] E. Michael, Continuous selections. I, Ann. of Math. 63 (1956), 361-382. Zbl0071.15902

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