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 φ : X → E a lower hemicontinuous multivalued map with nonempty convex closed values. Then there exists a continuous selection f : X → E of φ.

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.

Other selection theorems

• Zero-dimensional Michael selection theorem
• Aumann measurable selection theorem
• Bressan-Colombo directionally continuous selection theorem
• Castaing representation theorem
• Fryszkowski decomposable map selection
• Helly's selection theorem
• Kuratowski, Ryll-Nardzewski measurable selection theorem

References

• 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 
• 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