nLab
selection theorem

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

Selection theory asks if given a multi-valued function F:XYF\colon X \to Y does there exist a continuous selector f:XYf\colon X \to Y, i.e. a single-valued continuous function such that f(x)F(x)f(x) \in F(x) for all xXx \in X. A selection theorem states that under certain assumptions on X,Y,FX, Y, F there is indeed a selector. Normally, such assumptions include that XX is paracompact, YY is some subset of a topological vector space, FF is a lower semicontinuous map (also called hemicontinuous), and F(x)F(x) is convex for each xXx \in X.

More generally, one may ask if there is a multi-valued function G:XYG\colon X \to Y such that G(x)F(y)G(x) \subset F(y) for all yXy\in X and GG nicer behaved than FF, e.g. GG lower semicontinuous and G(x)G(x) compact for every xGx\in G or GG single-valued and measurable.

Selection theorems

Theorem

[Michael selection theorem] Let XX be paracompact, YY a Banach space, FF a lower semicontinuous map, and F(x)F(x) nonempty, convex, and closed for every xXx\in X. Then FF admits a selector.

References

Michael selection theorem appeared in

Overviews of selection theorems is found in

  • Dušan Repovš, Pavel V. Semenov, Continuous Selections of Multivalued Mappings. Dordrecht: Kluwer Academic Publishers 1998. ISBN 0-7923-5277-7.
  • Dušan Repovš, Pavel V. Semenov (2014). “Continuous Selections of Multivalued Mappings”, In Hart, K. P.; van Mill, J.; Simon, P. (eds.). Recent Progress in General Topology. III. Berlin: Springer. pp. 711–749. arXiv:1401.2257.

See also

Last revised on June 1, 2020 at 18:55:40. See the history of this page for a list of all contributions to it.