Closure of a set

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$A$ in a topological space $X$

The intersection of all closed sets of $X$ containing the set $A$.


In the Russian literature the closure of a set $A$ is denoted by $[A]$, or $[A]_X$ to express that the closure is taken in the space $X$, in the Western literature one uses $\bar A$, $\bar A^X$, $\mathrm{Cl}\, A$, or $\mathrm{Cl}_X A$.

Another definition of closure is as follows. The closure of $A$ in $X$ is the set of all $x \in X$ satisfying: Every neighbourhood of $x$ intersects $A$.

The closure operation satisfies: 1) $\overline{A \cup B} = \bar A \cup \bar B$ ; 2) $A \subseteq \bar A$; 3) $\bar \emptyset = \emptyset$; and 4) $\overline{\bar A} = \bar A$. Any operation satisfying 1), 2), 3), and 4) is called a closure operation. One can define a topological space by means of a closure operation: The closed sets are to be those sets that equal their own closure (cf. Closure relation).

This approach is taken in [a1].


[a1] K. Kuratowski, "Topology" , 1 , PWN & Acad. Press (1966) (Translated from French)
How to Cite This Entry:
Closure of a set. Encyclopedia of Mathematics. URL:
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