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

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.

The function: ${\displaystyle F(x)=[1-x/2,~1-x/4]}$, 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: ${\displaystyle f(x)=1-x/2}$ or ${\displaystyle f(x)=1-3x/8}$.

The function

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

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.²

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

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

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.

A theorem due to Deutsch and Kenderov generalizes Michel selection theorem to an equivalence relating approximate selections to almost lower hemicontinuity, where ${\displaystyle F}$ is said to be almost lower hemicontinuous if at each ${\displaystyle x\in X}$, all neighborhoods ${\displaystyle V}$ of ${\displaystyle 0}$ there exists a neighborhood ${\displaystyle U}$ of ${\displaystyle x}$ such that ${\displaystyle \cap _{u\in U}\{F(u)+V\}\neq \emptyset .}$

Precisely, Deutsch–Kenderov theorem states that if ${\displaystyle X}$ is paracompact, ${\displaystyle Y}$ a normed vector space and ${\displaystyle F(x)}$ is nonempty convex for each ${\displaystyle x\in X}$, then ${\displaystyle F}$ is almost lower hemicontinuous if and only if ${\displaystyle F}$ has continuous approximate selections, that is, for each neighborhood ${\displaystyle V}$ of ${\displaystyle 0}$ in ${\displaystyle Y}$ there is a continuous function ${\displaystyle f\colon X\mapsto Y}$ such that for each ${\displaystyle x\in X}$, ${\displaystyle f(x)\in F(X)+V}$.³

In a note Xu proved that Deutsch–Kenderov theorem is also valid if ${\displaystyle Y}$ is a locally convex topological vector space.⁴