# Difference between revisions of "Closed set"

From Encyclopedia of Mathematics

(Importing text file) |
(TeX) |
||

Line 1: | Line 1: | ||

+ | {{TEX|done}} | ||

''in a topological space'' | ''in a topological space'' | ||

− | A set containing all its limit points (cf. [[Limit point of a set|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 | + | A set containing all its limit points (cf. [[Limit point of a set|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==== | ====References==== | ||

<table><TR><TD valign="top">[1]</TD> <TD valign="top"> K. Kuratowski, "Topology" , '''1''' , PWN & Acad. Press (1966) (Translated from French)</TD></TR></table> | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> K. Kuratowski, "Topology" , '''1''' , PWN & Acad. Press (1966) (Translated from French)</TD></TR></table> |

## Latest revision as of 10:36, 16 April 2014

*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://encyclopediaofmath.org/index.php?title=Closed_set&oldid=16487

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