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: 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:, 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.

A function that does not satisfy lower hemicontinuity

The function 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 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 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 such that for each x \in X,. In a note Xu proved that Deutsch–Kenderov theorem is also valid if Y is a locally convex topological vector space.