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.

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

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

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 $$\cap_{u\in U} \{F(u)+V\} \ne \emptyset. $$

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.