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Closed set

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in a topological space

A set containing all its limit points (cf. Limit point of a set). Thus, all points of the complement to a closed set are interior points, and so a closed set can be defined as the complement to an open set. The concept of a closed set is basic to the definition of a topological space as a non-empty set $X$ with a distinguished system of sets (called closed sets) satisfying the following axioms: $X$ itself and the empty set $\emptyset$ are closed; the intersection of any number of closed sets is closed; the union of finitely many closed sets is closed.

References

[1] K. Kuratowski, "Topology" , 1 , PWN & Acad. Press (1966) (Translated from French)
How to Cite This Entry:
Closed set. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Closed_set&oldid=31773
This article was adapted from an original article by A.A. Mal'tsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article