Let $F$ and $G$ be two set valued maps from $X$ to $Y$. Assume that $F$ is closed, $G(x)$ is compact and $G$ is upper semi continuous at $x$. Then show that $F \cap G$ is upper semi continuous at $x$.
I know the definition of upper semi continuity, but I am not able to prove the above fact.
Please help.
(Also, can anyone please mention any book where set valued maps and their properties are explained in details?)
I dunno how to answer your main question, but for your question about book, there's book that explains set valued map in complete ways. That book is Set Valued Maps by Aubin-Celina