According to the wiki of [Kakutani's fixed-point theorem](http://en.wikipedia.org/wiki/Kakutani_fixed-point_theorem), A set-valued mapping $\varphi$ from a topological space $X$ into a powerset $\wp(Y)$ called upper semi-continuous if for every open set $W \subseteq Y$, $\lbrace x| \varphi(x) \subseteq W \rbrace$ is an open set in $X$.


My question:


1. What is the definition of continuity of a multi valued map $\varphi$?
2. What's the definition of open sets in $\wp(Y)$, in other words, what topology does $\wp(Y)$ have?



What is the definition of continuity of set-valued functions?
$\phi$ is upper semicontinuous if, for every open $W\subset Y$, the set $\lbrace x | \phi(x)\subset W\rbrace $ is open in $X$.


$\phi$ is lower semicontinuous if, for every open $W\subset Y$, the set $\lbrace x | \phi(x)\cap W\neq \emptyset\rbrace$ is open in $X$.


$\phi$ is continuous if it is both upper semincontinuous and lower semicontinuous. 


