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:[1] Michael Selection Theorem — Let X be a paracompact space and Y be a separable Banach space.

be a lower hemicontinuous set-valued function with nonempty convex closed values.

Then there exists a continuous selection

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.

, 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:

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

A theorem due to Deutsch and Kenderov generalizes Michel selection theorem to an equivalence relating approximate selections to almost lower hemicontinuity, where

there exists a neighborhood

Precisely, Deutsch–Kenderov theorem states that if

a normed vector space and

is nonempty convex for each

has continuous approximate selections, that is, for each neighborhood

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

is a locally convex topological vector space.