Michael selection theorem

Short description: On the existence of a continuous selection of a multivalued map from a paracompact space

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 [math]\displaystyle{ F\colon X\to Y }[/math] be a lower hemicontinuous set-valued function with nonempty convex closed values.

Then there exists a continuous selection [math]\displaystyle{ f\colon X \to Y }[/math] 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: [math]\displaystyle{ F(x)= [1-x/2, ~1-x/4] }[/math], 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: [math]\displaystyle{ f(x)= 1-x/2 }[/math] or [math]\displaystyle{ f(x)= 1-3x/8 }[/math].

A function that does not satisfy lower hemicontinuity

The function

[math]\displaystyle{ F(x)= \begin{cases} 3/4 & 0 \le x \lt 0.5 \\ \left[0,1\right] & x = 0.5 \\ 1/4 & 0.5 \lt x \le 1 \end{cases} }[/math]

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

[math]\displaystyle{ \frac{dx}{dt}(t)\in F(t,x(t)), \quad x(t_0)=x_0 }[/math]

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

Precisely, Deutsch–Kenderov theorem states that if [math]\displaystyle{ X }[/math] is paracompact, [math]\displaystyle{ Y }[/math] a normed vector space and [math]\displaystyle{ F (x) }[/math] is nonempty convex for each [math]\displaystyle{ x \in X }[/math], then [math]\displaystyle{ F }[/math] is almost lower hemicontinuous if and only if [math]\displaystyle{ F }[/math] has continuous approximate selections, that is, for each neighborhood [math]\displaystyle{ V }[/math] of [math]\displaystyle{ 0 }[/math] in [math]\displaystyle{ Y }[/math] there is a continuous function [math]\displaystyle{ f \colon X \mapsto Y }[/math] such that for each [math]\displaystyle{ x \in X }[/math], [math]\displaystyle{ f (x) \in F (X) + V }[/math].

In a note Xu proved that Deutsch–Kenderov theorem is also valid if [math]\displaystyle{ Y }[/math] 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.. Recent Progress in General Topology. III. Berlin: Springer. pp. 711–749. ISBN 978-94-6239-023-2. Bibcode: 2014arXiv1401.2257R. 
• Aubin, Jean-Pierre; Cellina, Arrigo (1984). Differential Inclusions, Set-Valued Maps And Viability Theory. Grundl. der Math. Wiss.. 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. 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. I. Kluwer. ISBN 0-7923-4682-3.