Closed map

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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.

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