View source for Closed set

''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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022580/c0225801.png" /> with a distinguished system of sets (called closed sets) satisfying the following axioms: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022580/c0225802.png" /> itself and the empty set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c022/c022580/c0225803.png" /> are closed; the intersection of any number of closed sets is closed; the union of finitely many closed sets is closed.

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