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:

1. What condition required $\varphi$ is continuous?
2. What's the definition of open sets in $\wp(Y)$, in other words, what topo does $\wp(Y)$ have?

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:

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?

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 $W \subseteq Y$, $\{x| \varphi(x) \subseteq W\}$ is an open set in $X$.

My question:

1. What condition required $\varphi$ is continuous?
2. What's the definition of open sets in $\wp(Y)$, in other words, what topo does $\wp(Y)$ have?

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:

1. What condition required $\varphi$ is continuous?
2. What's the definition of open sets in $\wp(Y)$, in other words, what topo does $\wp(Y)$ have?