Michael selection theorem explained

In functional analysis, a branch of mathematics, Michael selection theorem is a selection theorem named after Ernest Michael. In its most popular form, it states the following:^[1]

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

Examples

A function that satisfies all requirements

The function:

F(x) = [ 1- x/ 2 , ~ 1- x/ 4 ] , shown by the grey area in the figure at the right, is a set-valued function from the real interval [0,1] to itself. It satisfies all Michael's conditions, and indeed it has a continuous selection, for example: f(x) = 1- x/ 2 or f(x) = 1-3 x/ 8 .

A function that does not satisfy lower hemicontinuity

The function

F(x) = \begin{cases} 3 / 4 & 0 \le x < 0 . 5 \ \left[ 0 , 1 \right] & x = 0 . 5 \ 1 / 4 & 0 . 5 < x \le 1 \end{cases}

is a set-valued function from the real interval [0,1] to itself. It has nonempty convex closed values. However, it is not lower hemicontinuous at 0.5. Indeed, Michael's theorem does not apply and the function does not have a continuous selection: any selection at 0.5 is necessarily discontinuous.^[2]

Applications

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

(dx)/(dt) (t)\in F(t,x(t)), x(t _{0)=x} _{0}

has a C¹ 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 an equivalence relating approximate selections to almost lower hemicontinuity, where

F is said to be almost lower hemicontinuous if at each x \in X , all neighborhoods V of 0 there exists a neighborhood U of x such that \cap _{u\in} \{F(u) + V\} \ne \emptyset.

Precisely, Deutsch–Kenderov theorem states that if

X is paracompact, Y a normed vector space and F (x) is nonempty convex for each x \in 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 Y there is a continuous function f \colon X \mapsto Y such that for each x \in X , f (x) \in F (X) + V .^[3]

In a note Xu proved that Deutsch–Kenderov theorem is also valid if

Y is a locally convex topological vector space.^[4]

See also

• Zero-dimensional Michael selection theorem
• Selection theorem
• Maximum theorem

Further reading

• Book: Dušan . Repovš . Dušan Repovš. Pavel V. . Semenov . Continuous Selections of Multivalued Mappings . Hart . K. P. . van Mill . J. . Simon . P. . Recent Progress in General Topology . III . 2014 . Springer . Berlin . 978-94-6239-023-2 . 711–749 . 1401.2257 . 2014arXiv1401.2257R .
• Book: Aubin, Jean-Pierre . Arrigo . Cellina . Differential Inclusions, Set-Valued Maps And Viability Theory . Grundl. der Math. Wiss. . 264 . Springer-Verlag . Berlin . 1984 . 3-540-13105-1 .
• Book: Aubin, Jean-Pierre . H. . Frankowska . Hélène Frankowska. Set-Valued Analysis . Birkhäuser . Basel . 1990 . 3-7643-3478-9 .
• Book: Deimling, Klaus . Multivalued Differential Equations . Walter de Gruyter . 1992 . 3-11-013212-5 .
• Book: Dušan . Repovš . Dušan Repovš . Pavel V. . Semenov . Continuous Selections of Multivalued Mappings . Kluwer Academic Publishers . Dordrecht . 1998 . 0-7923-5277-7 .
• Dušan . Repovš . Dušan Repovš. Pavel V. . Semenov . Ernest Michael and Theory of Continuous Selections . . 155 . 8 . 2008 . 755–763 . 10.1016/j.topol.2006.06.011 . 0803.4473 .
• Book: Aliprantis, Charalambos D. . Kim C. . Border . Infinite Dimensional Analysis : Hitchhiker's Guide . Springer . 3rd . 2007 . 978-3-540-32696-0 .
• Book: Hu, S. . N. . Papageorgiou . Handbook of Multivalued Analysis . I . Kluwer . 0-7923-4682-3 .

Notes and References

1. Michael . Ernest . Ernest Michael . Continuous selections. I . 0077107 . 1956 . . Second Series . 63 . 361–382 . 2 . 1969615 . 10.2307/1969615. 10338.dmlcz/119700 . free .
2. Web site: proof verification - Reducing Kakutani's fixed-point theorem to Brouwer's using a selection theorem. Mathematics Stack Exchange. 2019-10-29.
3. Deutsch. Frank. Kenderov. Petar. Continuous Selections and Approximate Selection for Set-Valued Mappings and Applications to Metric Projections. SIAM Journal on Mathematical Analysis. January 1983. 14. 1. 185–194. 10.1137/0514015.
4. Xu. Yuguang. A Note on a Continuous Approximate Selection Theorem. Journal of Approximation Theory. December 2001. 113. 2. 324–325. 10.1006/jath.2001.3622. free.