Michael selection theorem

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.

Going deeper

For a student

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 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. Compare with the fixed-point theorems for upper hemicontinuous maps, such as Kakutani's, where the opposite kind of hemicontinuity is the right one.