# Closed set

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A set is closed if it contains all its limit points, i.e. if $LaTeX: x^k \rightarrow x$ and each $LaTeX: x^k$ is in $LaTeX: X$, then $LaTeX: x$ is in $LaTeX: X$. By the logic of the definition, an empty set is closed (it is also open). A finite number of intersections of closed sets is a closed set, so the feasibility region of a mathematical program in standard form is closed if each constraint function is continuous. ($LaTeX: g$ can be only upper semi-continuous for the constraint, $LaTeX: g(x) \le 0$.) An example of a set that is not closed is: $LaTeX: {x: x < 0}$. The point 0 can be approached by a sequence, say $LaTeX: x^k = 1/k$, but 0 is not in the set. The closure of a set is the union of the set and all of its limit points.