View source for Closure of a set

''$A$ in a topological space $X$''

The intersection of all [[closed set]]s of $X$ containing the set $A$.



====Comments====
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 [[#References|[a1]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Kuratowski,   "Topology" , '''1''' , PWN &amp; Acad. Press  (1966)  (Translated from French)</TD></TR></table>

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[[Category:General topology]]