# Michael selection theorem

In functional analysis, a branch of mathematics, the most popular version of the **Michael selection theorem**, named after Ernest Michael, states the following:

- Let
*E*be a Banach space,*X*a paracompact space and*F*:*X*→*E*a lower hemicontinuous multivalued map with nonempty convex closed values. Then there exists a continuous selection*f*:*X*→*E*of*F*.

- Conversely, if any lower semicontinuous multimap from topological space
*X*to a Banach space, with nonempty convex closed values admits continuous selection, then*X*is paracompact. This provides another characterization for paracompactness.

## Applications

Michael selection theorem can be applied to show that the differential inclusion

has a *C*^{1} solution when *F* is lower semi-continuous and *F*(*t*, *x*) is a nonempty closed and convex set for all (*t*, *x*). When *F* is single valued, this is the classic Peano existence theorem.

## Generalizations

A theorem due to Deutsch and Kenderov generalizes Michel selection theorem to a equivalence relating approximate selections to almost lower hemicontinuity, where *F* is said to be almost lower hemicontinuous if at each *x*∈*X*, all neighborhoods *V* of *0* there exists a neighborhood *U* of *x* such that
Precisely, Deutsch and Kenderov theorem states that if *X* is paracompact, *E* a normed vector space and *F*(*x*) is nonempty convex for each *x*∈*X*, then *F* is almost lower hemicontinuous if and only if *F* has continuous approximate selections, that is, for each neighborhood *V* of 0 in *E* there is a continuous function *f*:*X* → *E* such that for each *x* ∈ *X*, *f*(*x*) ∈ *F*(*X*) + *V*.^{[1]}

In a note of Y. Xu it is proved that Deutsch and Kenderov theorem is also valid if *E* is locally convex topological vector space.^{[2]}

## See also

- Zero-dimensional Michael selection theorem
- List of selection theorems

## References

- ↑ Deutsch, Frank; Kenderov, Petar (January 1983). "Continuous Selections and Approximate Selection for Set-Valued Mappings and Applications to Metric Projections".
*SIAM Journal on Mathematical Analysis*.**14**(1): 185–194. doi:10.1137/0514015. - ↑ Xu, Yuguang (December 2001). "A Note on a Continuous Approximate Selection Theorem".
*Journal of Approximation Theory*.**113**(2): 324–325. doi:10.1006/jath.2001.3622.

- Michael, Ernest (1956), "Continuous selections. I",
*Annals of Mathematics. Second Series*, Annals of Mathematics,**63**(2): 361–382, doi:10.2307/1969615, JSTOR 1969615, MR 0077107 - Dušan Repovš; Pavel V. Semenov (2014). "Continuous Selections of Multivalued Mappings". In Hart, K. P.; van Mill, J.; Simon, P.
*Recent progress in general topology III*. Berlin: Springer. pp. 711–749. ISBN 978-94-6239-023-2. - Jean-Pierre Aubin, Arrigo Cellina
*Differential Inclusions, Set-Valued Maps And Viability Theory*, Grundl. der Math. Wiss., vol. 264, Springer - Verlag, Berlin, 1984 - J.-P. Aubin and H. Frankowska
*Set-Valued Analysis*, Birkh¨auser, Basel, 1990 - Klaus Deimling
*Multivalued Differential Equations*, Walter de Gruyter, 1992 - D.Repovs and P. V. Semenov, Continuous Selections of Multivalued Mappings, Kluwer Academic Publishers, Dordrecht 1998.
- D.Repovs and P. V. Semenov, Ernest Michael and theory of continuous selections, Topol. Appl. 155:8 (2008), 755-763.
- Aliprantis, Kim C. Border
*Infinite dimensional analysis. Hitchhiker's guide*Springer - S.Hu, N.Papageorgiou
*Handbook of multivalued analysis. Vol. I*Kluwer