On the existence of a continuous selection of a multivalued map from a paracompact space
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]
Let X be a paracompact space and Y a Banach space.
Let be a lower hemicontinuous set-valued function with nonempty convex closed values.
Then there exists a continuous selection 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.
^Michael, Ernest (1956). "Continuous selections. I". Annals of Mathematics. Second Series. 63 (2): 361–382. doi:10.2307/1969615. hdl:10338.dmlcz/119700. JSTOR 1969615. MR 0077107.
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