---
tag: 3RJ1K36A9A
kind: statement
role: theorem
name: Michael selection theorem
msc: [54C65, 54C60]
uses: [8R29G6NG9Q, 0S1FJ8VEX9, 06GZ956657, 03CPD1HGF3, XNMKX7H4FD, 02AS45T3H9, 0ZCK8DQYN1, 0VGFHTM75Y]
about: [XNMKX7H4FD, 03CPD1HGF3]
sources:
  - url: https://en.wikipedia.org/wiki/Michael_selection_theorem
same_as:
  wikidata: Q6835566
  wikipedia: Michael selection theorem
---

## Statement

Let $X$ be a [[paracompact|8R29G6NG9Q]] [[Hausdorff space|0S1FJ8VEX9]] and let $Y$ be a [[Banach space|06GZ956657]].
Let $F$ be a [[multivalued function|03CPD1HGF3]] from $X$ to $Y$ such that:

1. $F$ is [[lower hemicontinuous|XNMKX7H4FD]].
2. For every $x \in X$, the set $F(x)$ is nonempty, [[convex|02AS45T3H9]] and [[closed|0ZCK8DQYN1]] in $Y$.

Then there is a [[continuous|0VGFHTM75Y]] map $f \colon X \to Y$ with $f(x) \in F(x)$ for every $x \in X$.

## Remarks

Such a map $f$ is a *continuous selection* of $F$.
The proof first builds, for each $\varepsilon > 0$, a continuous map that lies within distance $\varepsilon$ of $F(x)$ at every $x$.
It glues local choices with a partition of unity, which exists because $X$ is paracompact.
Convexity keeps the convex combinations close to $F(x)$.
Repeating this with $\varepsilon = 2^{-n}$, each step close to the previous one, gives a uniformly Cauchy sequence of continuous maps.
Its limit exists since $Y$ is complete, it is continuous, and it lies in $F(x)$ since $F(x)$ is closed.

Example: let $T \colon Y \to Z$ be a continuous linear surjection between Banach spaces and set $F(z) = T^{-1}(z)$.
The values are closed affine subspaces, and the open mapping theorem makes $F$ lower hemicontinuous.
So $T$ has a continuous, possibly nonlinear, right inverse.

Lower hemicontinuity cannot be replaced by upper hemicontinuity.
On $X = \R$, let $F(x) = \set{-1}$ for $x < 0$, $F(0) = [-1, 1]$ and $F(x) = \set{1}$ for $x > 0$.
This $F$ is upper hemicontinuous with nonempty closed convex values, but a selection must jump from $-1$ to $1$ at $0$, so no continuous selection exists.

## Explanations

### L1 human

Suppose that at every point of a space you are given a set of allowed values, and the sets change in a reasonable way.
The theorem says that you can choose one allowed value at each point so that the choice varies continuously.
This needs the allowed sets to be convex and closed, the space of values to be a complete normed space, and the sets not to shrink suddenly.

### L2 human

The theorem is the main tool to turn set-valued information into continuous functions.
It gives continuous right inverses of linear surjections between Banach spaces, and it is used in the theory of [[absolute neighborhood retracts|DK3MT3HYX5]] and in fixed-point and control theory.
Paracompactness enters only through partitions of unity, in the same way as in the proof of the [[Dugundji extension theorem|RBDJBK63HX]].
Compare with the fixed-point theorems for upper hemicontinuous maps, such as Kakutani's, where the opposite kind of hemicontinuity is the right one.