Regularity of the boundary of the graph of an set-valued map

Question

I'm very new to the topic. I was wondering the following: with $X\subset\mathbb R^n$ a compact subset, $n\in\mathbb N^+$ and $Y=\mathbb R^m$, $m\in\mathbb N^+$ consider a set-valued map $F: X\to 2^Y$. Assume that his graph is compact, is there anything I can say about the regularity of the boundary of $\operatorname{graph}(F)$? Do you have any sharp reference to this that are accessible to a non-mathematician?

Answer 1

I haven't seen such a construction before. As long as the topology on $2^Y$ is Hausdorff, then following argument works.

Assuming that the topology on $2^Y$ is Hausdorff, then $X\times 2^Y$ is Hausdorff. As a compact subset of $X\times 2^Y$, we know that the graph of $F$, denoted by $\Gamma(F)$, is closed. Therefore it contains its boundary $\partial\Gamma(F)$. This means that $\partial\Gamma(F)$ a closed subset of a compact Hausdorff space. Hence $\partial\Gamma(F)$ is compact and Hausdorff, and consequently it is regular.