Closure - Revision history

GregG: fixup

2017-01-07T04:19:45Z

fixup

2008-06-28T03:34:10Z

links

2008-06-13T22:06:20Z

added abstract non-topological version.

2008-06-13T18:34:48Z

2008-06-13T18:34:41Z

2007-04-07T01:58:54Z

New page: The '''closure''' of a set ''A'' is the intersection of all closed sets containing ''A''. Category:Topology

New page

The '''closure''' of a set ''A'' is the intersection of all [[closed set]]s containing ''A''.

[[Category:Topology]]

@@ Line 1: / Line 1: @@
-In [[topology]], the '''closure''' of a set ''A'' is the intersection of all [[closed set]]s containing ''A''. In [[metric space]]s, this can be also defined as the set of all [[limit point]]s of the set ''A''.
+In [[topology]], the '''closure''' of a set <math>A</math> is the intersection of all [[closed set]]s containing <math>A</math>. Equivalently, the closure of <math>A</math> is the union of <math>A</math> and all [[limit point]]s of <math>A</math>
-A '''closure operator''' is an abstract ([[category theory]]) form of the topological notion of closure which can be applied to any set ''A''. It is a [[function]] ''cl'' from ''A'' to the [[power set]] of ''A'' satisfying the following conditions:
+A '''closure operator''' is an abstract ([[category theory]]) form of the topological notion of closure which can be applied to any set <math>A</math>. It is a [[function]] <math>cl</math> from <math>A</math> to the [[power set]] of <math>A</math> satisfying the following conditions:
 * <math>A\in cl(A)</math> ([[augment]]ation)

@@ Line 3: / Line 3: @@
 A '''closure operator''' is an abstract ([[category theory]]) form of the topological notion of closure which can be applied to any set ''A''. It is a [[function]] ''cl'' from ''A'' to the [[power set]] of ''A'' satisfying the following conditions:
-* <math>A\in cl(A)</math> (augmentation)
+* <math>A\in cl(A)</math> ([[augment]]ation)
-* If <math>B\subset A</math> then <math>cl(B)\subset cl(A)</math> (monotonicity)
+* If <math>B\subset A</math> then <math>cl(B)\subset cl(A)</math> ([[Monotone|monotonicity]])
-* <math>cl(A)\in cl(A)</math> (idempotence)
+* <math>cl(A)\in cl(A)</math> ([[idempotent|idempotence]])
 In the category of [[topological space]]s, this operator is isomorphic to the standard topological one.
 [[Category:Topology]]

← Older revision		Revision as of 22:06, June 13, 2008
Line 1:		Line 1:
−	~~The~~ '''closure''' of a set ''A'' is the intersection of all [[closed set]]s containing ''A''. In [[metric space]]s, this can be also defined as the set of all limit ~~points~~ of the set ''A''.	+	In [[topology]], the '''closure''' of a set ''A'' is the intersection of all [[closed set]]s containing ''A''. In [[metric space]]s, this can be also defined as the set of all [[limit point]]s of the set ''A''.
		+
		+	A '''closure operator''' is an abstract ([[category theory]]) form of the topological notion of closure which can be applied to any set ''A''. It is a [[function]] ''cl'' from ''A'' to the [[power set]] of ''A'' satisfying the following conditions:
		+
		+	* <math>A\in cl(A)</math> (augmentation)
		+	* If <math>B\subset A</math> then <math>cl(B)\subset cl(A)</math> (monotonicity)
		+	* <math>cl(A)\in cl(A)</math> (idempotence)
		+
		+	In the category of [[topological space]]s, this operator is isomorphic to the standard topological one.

	[[Category:Topology]]		[[Category:Topology]]

← Older revision		Revision as of 18:34, June 13, 2008
Line 1:		Line 1:
−	The '''closure''' of a set ''A'' is the intersection of all [[closed set]]s containing ''A''. In metric ~~spaces~~, this can be also defined as the set of all limit points of the set ''A''.	+	The '''closure''' of a set ''A'' is the intersection of all [[closed set]]s containing ''A''. In [[metric space]]s, this can be also defined as the set of all limit points of the set ''A''.

	[[Category:Topology]]		[[Category:Topology]]

← Older revision		Revision as of 18:34, June 13, 2008
Line 1:		Line 1:
−	The '''closure''' of a set ''A'' is the intersection of all [[closed set]]s containing ''A''.	+	The '''closure''' of a set ''A'' is the intersection of all [[closed set]]s containing ''A''. In metric spaces, this can be also defined as the set of all limit points of the set ''A''.

	[[Category:Topology]]		[[Category:Topology]]