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 ${\displaystyle F\colon X\to Y}$ be a lower hemicontinuous set-valued function with nonempty convex closed values.

Then there exists a continuous selection ${\displaystyle f\colon X\to Y}$ 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.