What is the definition of continuity of set-valued functions? According to the wiki of Kakutani's 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:


*

*What is the definition of continuity of a multi valued map  $\varphi$?

*What's the definition of open sets in $\wp(Y)$, in other words, what topology does $\wp(Y)$ have?

 A: The definition quoted is an "order" notion of upper semicontinuous, not a "topology" notion.  For real-valued functions, the two coincide.  But in other settings you can have one but not the other.
A: $\phi$ 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.  
A: One sensible way of generalizing continuity to set-valued functions (from $X$ to subsets of $Y$) is to require the graph of the function to be closed in the product $X\times Y$. This would be equivalent to the continuity of the function if $Y$ is compact.  Thus, the Heaviside function is not continuous because one of the points 0 or 1 on the $y$-axis is not in the graph,  but if one redefines it to take both values at 0, the graph becomes closed subset of the plane.  See http://en.wikipedia.org/wiki/Closed_graph_theorem for a related (but different) notion.
