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)=
-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 ,1to 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
3/4 & 0 \le x < 0.5 \\
\left ,1\right & x = 0.5 \\
1/4 & 0.5 < x \le 1
\end

is a set-valued function from the real interval ,1to 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

:$\frac(t)\in F(t,x(t)), \quad x(t_0)=x_0$

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 $\cap_ \ \ne \emptyset.$

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.

See also

* Zero-dimensional Michael selection theorem
*Selection theorem

References

Further reading

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{{Functional analysis

Theory of continuous functions
Properties of topological spaces
Theorems in functional analysis
Compactness theorems