\subset V$. We say $f$ is upper semi-continuous if $f$ is upper semi-continuous at all $x \in X$. If $f \colon X \rightarrow Cl(Y)$, the graph of $f$ (denoted $G(f)$) is the set of all $(x,y) \in X \times Y$ such that $y \in f(x)$. Recall that a topological space being