Separable space - Revision history

Vipul: /* Relation with other properties */

2012-01-24T21:35:55Z

Relation with other properties

← Older revision		Revision as of 21:35, 24 January 2012
Line 21:		Line 21:
	\|-		\|-
	\| [[Weaker than::hereditarily separable space]] \|\| \|\| \|\| \|\|		\| [[Weaker than::hereditarily separable space]] \|\| \|\| \|\| \|\|
			\|-
			\| [[Weaker than::sub-Euclidean space]] \|\| \|\| \|\| \|\|
			\|-
			\| [[Weaker than::manifold]] \|\| \|\| \|\| \|\|
	\|}		\|}

Vipul: /* Relation with other properties */

2012-01-24T21:34:07Z

Relation with other properties

← Older revision		Revision as of 21:34, 24 January 2012
Line 13:		Line 13:
	===Stronger properties===		===Stronger properties===

	* [[~~Second~~-countable space]]		{\| class="sortable" border="1"
	* [[Polish space]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[~~Hereditarily~~ separable space]]		\|-
			\| [[Weaker than::second-countable space]] \|\| \|\| \|\| \|\|
			\|-
			\| [[Weaker than::Polish space]] \|\| \|\| \|\| \|\|
			\|-
			\| [[Weaker than::hereditarily separable space]] \|\| \|\| \|\| \|\|
			\|}

	==Metaproperties==		==Metaproperties==

Vipul: 6 revisions

2008-05-11T19:58:02Z

6 revisions

← Older revision	Revision as of 19:58, 11 May 2008
(No difference)

Vipul at 22:13, 21 April 2008

2008-04-21T22:13:38Z

← Older revision		Revision as of 22:13, 21 April 2008
Line 34:		Line 34:

	{{applyingoperatorgives\|hereditarily operator\|hereditarily separable space}}		{{applyingoperatorgives\|hereditarily operator\|hereditarily separable space}}

			==References==
			===Textbook references===
			* {{booklink\|Munkres}}, Page 192 (definition in paragraph)

Vipul at 22:12, 15 December 2007

2007-12-15T22:12:10Z

← Older revision		Revision as of 22:12, 15 December 2007
Line 30:		Line 30:

	It is not true in general that a subspace of a separable space is separable. A counterexample is the antidiagonal in the [[Sorgenfrey plane]].		It is not true in general that a subspace of a separable space is separable. A counterexample is the antidiagonal in the [[Sorgenfrey plane]].

			==Effect of property operators==

			{{applyingoperatorgives\|hereditarily operator\|hereditarily separable space}}

Vipul at 19:43, 15 December 2007

2007-12-15T19:43:32Z

← Older revision		Revision as of 19:43, 15 December 2007
Line 15:		Line 15:
	* [[Second-countable space]]		* [[Second-countable space]]
	* [[Polish space]]		* [[Polish space]]
			* [[Hereditarily separable space]]

			==Metaproperties==

			{{finite-DP-closed}}

			A finite product of separable spaces is separable. The countable dense subset that we take for the product is the Cartesian product of countable dense subsets for each.

			{{open subspace-closed}}

			Any [[open subset]] of a separable space is separable. Indeed, given a countable dense subset for the whole space, we intersect it with the open subset to get a countable dense subset of the open subset.

			{{not subspace-closed}}

			It is not true in general that a subspace of a separable space is separable. A counterexample is the antidiagonal in the [[Sorgenfrey plane]].

Vipul: /* Relation with other properties */

2007-12-11T20:14:19Z

Relation with other properties

← Older revision		Revision as of 20:14, 11 December 2007
Line 14:		Line 14:

	* [[Second-countable space]]		* [[Second-countable space]]
			* [[Polish space]]

Vipul at 18:27, 26 October 2007

2007-10-26T18:27:49Z

@@ Line 1: / Line 1: @@
 {{topospace property}}
 ==Definition==
@@ Line 6: / Line 8: @@
 A [[topological space]] is said to be '''separable''' if it has a countable [[dense subset]].

Vipul at 08:56, 18 August 2007

2007-08-18T08:56:59Z

New page

{{topospace property}}

==Definition==

===Symbol-free definition===

A [[topological space]] is said to be '''separable''' if it has a countable [[dense subset]].