Michael selection theorem
# 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 $C^1$ 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.

