nLab selection theorem

Idea

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

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

Selection theorems

Theorem

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

Related entries

• multi-valued function
• Michael's theorem

References

Michael selection theorem appeared in

• Ernest Michael, Continuous selections. I, Annals of Math. 63 (2): 361–382 (doi:10.2307/1969615, MR0077107)

Overviews of selection theorems is found in

• Dušan Repovš, Pavel V. Semenov, Continuous Selections of Multivalued Mappings. Dordrecht: Kluwer Academic Publishers 1998. ISBN 0-7923-5277-7.
• Dušan Repovš, Pavel V. Semenov (2014). “Continuous Selections of Multivalued Mappings”, In Hart, K. P.; van Mill, J.; Simon, P. (eds.). Recent Progress in General Topology. III. Berlin: Springer. pp. 711–749. arXiv:1401.2257.

See also

• Wikipedia, Selection theorem