selection theorem



topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory


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Analysis Theorems

topological homotopy theory



Selection theory asks if given a multi-valued function F:XYF\colon X \to Y does there exist a continuous selector f:XYf\colon X \to Y, i.e. a single-valued continuous function such that f(x)F(x)f(x) \in F(x) for all xXx \in X. A selection theorem states that under certain assumptions on X,Y,FX, Y, F there is indeed a selector. Normally, such assumptions include that XX is paracompact, YY is some subset of a topological vector space, FF is a lower semicontinuous map (also called hemicontinuous), and F(x)F(x) is convex for each xXx \in X.

More generally, one may ask if there is a multi-valued function G:XYG\colon X \to Y such that G(x)F(y)G(x) \subset F(y) for all yXy\in X and GG nicer behaved than FF, e.g. GG lower semicontinuous and G(x)G(x) compact for every xGx\in G or GG single-valued and measurable.

Selection theorems


[Michael selection theorem] Let XX be paracompact, YY a Banach space, FF a lower semicontinuous map, and F(x)F(x) nonempty, convex, and closed for every xXx\in X. Then FF admits a selector.


Michael selection theorem appeared in

Overviews of selection theorems is found in

See also