# Closed map

Let $LaTeX: A: X \rightarrow 2^X$ be a point-to-set map and suppose $LaTeX: x^k \rightarrow x$ and $LaTeX: y^k \rightarrow y$ are such that $LaTeX: y^k \in A(x^k)$ for all $LaTeX: k$. Then, $LaTeX: A$ is closed at $LaTeX: x$ if $LaTeX: y \in A(x)$. The map $LaTeX: A$ is closed on a subset of $LaTeX: X$, say $LaTeX: S$, if $LaTeX: A$ is closed at each $LaTeX: x$ in $LaTeX: S$.