Given two sets X and Y, let F be a set-valued function from X and Y. Equivalently, is a function from X to the power set of Y.
A function is said to be a selection of F if
In other words, given an input x for which the original function F returns multiple values, the new function f returns a single value. This is a special case of a choice function.
The axiom of choice implies that a selection function always exists; however, it is often important that the selection have some "nice" properties, such as continuity or measurability. This is where the selection theorems come into action: they guarantee that, if F satisfies certain properties, then it has a selection f that is continuous or has other desirable properties.
for all x in X, the set F(x) is nonempty, convex and closed.
The approximate selection theorem[3] states the following:
Suppose X is a compact metric space, Y a non-empty compact, convex subset of a normed vector space, and Φ: X → a multifunction all of whose values are compact and convex. If graph(Φ) is closed, then for every ε > 0 there exists a continuous function f : X → Y with graph(f) ⊂ [graph(Φ)]ε.
Here, denotes the -dilation of , that is, the union of radius- open balls centered on points in . The theorem implies the existence of a continuous approximate selection.
Another set of sufficient conditions for the existence of a continuous approximate selection is given by the Deutsch–Kenderov theorem,[4] whose conditions are more general than those of Michael's theorem (and thus the selection is only approximate):
^Deutsch, Frank; Kenderov, Petar (January 1983). "Continuous Selections and Approximate Selection for Set-Valued Mappings and Applications to Metric Projections". SIAM Journal on Mathematical Analysis. 14 (1): 185–194. doi:10.1137/0514015.