Michael selection theorem

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:

Let X be a paracompact space and Y a Banach space.

Let $F\colon X\to Y$ be a lower hemicontinuous set-valued function with nonempty convex closed values.

Then there exists a continuous selection $f\colon X\to Y$ of F.

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 $f(x)=1-3x/8$.

A function that does not satisfy lower hemicontinuity

The function

$F(x)={\begin{cases}3/4&0\leq x<0.5\\\left[0,1\right]&x=0.5\\1/4&0.5<x\leq 1\end{cases}}$

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.

Applications

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

${\frac {dx}{dt}}(t)\in F(t,x(t)),\quad x(t_{0})=x_{0}$

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 $\cap _{u\in U}\{F(u)+V\}\neq \emptyset .$

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 $x\in X$, $f(x)\in F(X)+V$.

In a note Xu proved that Deutsch–Kenderov theorem is also valid if $Y$ is a locally convex topological vector space.

See also

• Zero-dimensional Michael selection theorem
• Selection theorem

Further reading

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