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.


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