# Selections and representations of multifunctions in paracompact spaces

Alberto Bressan; Giovanni Colombo

Studia Mathematica (1992)

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

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topBressan, 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

top- [1] J. P. Aubin and A. Cellina, Differential Inclusions, Springer, Berlin 1984.
- [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.
- [5] A. Bressan and G. Colombo, Boundary value problems for lower semicontinuous differential inclusions, Funkcial. Ekvac., to appear. Zbl0788.34007
- [6] A. Bressan and A. Cortesi, Directionally continuous selections in Banach spaces, Nonlin. Anal. 13 (1989), 987-992. Zbl0687.34013
- [7] E. Michael, Selected selection theorems, Amer. Math. Monthly 63 (1956), 233-238. Zbl0070.39502
- [8] E. Michael, Continuous selections. I, Ann. of Math. 63 (1956), 361-382. Zbl0071.15902

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