I am currently working on a continuous selector problem of multifunctions. I am trying to figure out if a continuous multifunction defined on a compact metric space always admit a continuous selector.
A mapping $\phi$ from a topological space $X$ to the family of subsets of a topological space $Y$ is called multifunction. It is upper hemicontinuous if the set $\{ x \in X : \phi (x) \subset U \}$ is open in $X$ for every open set of $Y$. It is lower hemicontinuous if $ \{ x \in X : \phi (x) \cap U \not = \emptyset \}$ is open in $X$ for every open set of $Y$. It is continuous if it is upper and lower hemicontinuous. A function $f: X \to Y$ is a continuous selector of $\phi$ if $f$ is continuous and $f(x) \in \phi(x)$ for all $x \in X.$
Do you know of any examples of continuous multifunctions defined on a compact metric space that do not have a continuous selection?
I know the following Michael selection theorem:
A lower hemicontinuous multifunction with nonempty and convex values from a paracompact space into a completely metrizable locally convex space admits a continuous selector.
I am wondering if it is possible that a continuous multifunction does not have a continuous selector if the multifunction is not convex-valued.
I would greatly appreciate insights or suggestions.
Thank you.
