Need reference to continuity of inverse multivalued function.

Question

Let's say $X$ and $Y$ are compact metric spaces. Also, multivalued function $f : X \rightarrow 2^Y$ is continuous in Hausdorff sense. I'm sure it is proven somewhere, that in this case $f^{-1} : Y \rightarrow 2^X$ such as $f^{-1}(y) = \{x \in X \mid y \in f(x)\}$ is continuous too. Can anyone provide reference for this result?

Answer 1

The Hausdorff metric topology is the same as the Vietoris topology on $2^X$ (or $2^Y$) when $X$ ($Y$) is compact metric.

The standard subbase for this are all sets of the form (where $U$ is non-empty open in $X$):

$$\langle U\rangle =\{F \in 2^X: F \cap U \neq \emptyset\} \text{ , and } [U]=\{F \in 2^X: F \subseteq U\}\text{.}$$

Note that the sets $[F]$ and $\langle F \rangle$ are closed in $2^X$ for $F \subseteq X$ closed, as $$2^X \setminus [F] = \langle X\setminus F \rangle$$ and

$$2^X \setminus \langle F \rangle = [X \setminus F]$$

both by definition.

Now, we are given $f: X \to 2^Y$ which is continuous, and we define

$F: Y \to 2^X$ by $F(y)=\{x \in X: y \in f(x)\}$. This set can be written as $f^{-1}[[\{y\}]]$, preimage under $f$ of a closed set of $2^Y$ and so closed by continuity of $f$; hence the map is well-defined, also note $f(x)$ always being non-empty implies the same for $F(y)$.

Now compute $F^{-1}[[U]]$ and $F^{-1}[\langle U \rangle]$ and check these are open.