Difference between revisions of "Closure of a set"
From Encyclopedia of Mathematics
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| − | ''in a topological space'' | + | ''$A$ in a topological space $X$'' |
| − | The intersection of all | + | The intersection of all [[closed set]]s of $X$ containing the set $A$. |
====Comments==== | ====Comments==== | ||
| − | In the Russian literature the closure of a set | + | 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 | + | 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) | + | 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]]]. | This approach is taken in [[#References|[a1]]]. | ||
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====References==== | ====References==== | ||
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Kuratowski, "Topology" , '''1''' , PWN & Acad. Press (1966) (Translated from French)</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> K. Kuratowski, "Topology" , '''1''' , PWN & Acad. Press (1966) (Translated from French)</TD></TR></table> | ||
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| + | {{TEX|done}} | ||
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| + | [[Category:General topology]] | ||
Latest revision as of 16:57, 9 November 2014
$A$ in a topological space $X$
The intersection of all closed sets 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 [a1].
References
| [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: http://encyclopediaofmath.org/index.php?title=Closure_of_a_set&oldid=13149
Closure of a set. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Closure_of_a_set&oldid=13149
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