Sequential space - Revision history

Vipul: 6 revisions

2008-05-11T19:58:10Z

6 revisions

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

Vipul: /* Stronger properties */

2007-12-18T18:17:04Z

Stronger properties

← Older revision		Revision as of 18:17, 18 December 2007
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	* [[Radial space]]		* [[Radial space]]
	* [[First-countable space]]		* [[First-countable space]]
			* [[Metrizable space]]
			* [[Locally metrizable space]]

	==References==		==References==

Vipul: /* Stronger properties */

2007-12-17T19:30:24Z

Stronger properties

← Older revision		Revision as of 19:30, 17 December 2007
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	* [[Radial space]]		* [[Radial space]]
	* [[Limit point-compact space]]
	* [[First-countable space]]		* [[First-countable space]]

Vipul: /* Stronger properties */

2007-12-17T19:29:45Z

Stronger properties

← Older revision		Revision as of 19:29, 17 December 2007
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	* [[Radial space]]		* [[Radial space]]
			* [[Limit point-compact space]]
	* [[First-countable space]]		* [[First-countable space]]

Vipul: /* Relation with oher properties */

2007-12-17T19:29:26Z

Relation with oher properties

← Older revision		Revision as of 19:29, 17 December 2007
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	Here, a sequentially closed subset is a subset that contains the limit of every convergent sequence in it.		Here, a sequentially closed subset is a subset that contains the limit of every convergent sequence in it.

	==Relation with ~~oher~~ properties==		==Relation with other properties==

	===Stronger properties===		===Stronger properties===

	* [[~~Frechet-Urysohn~~ space]] ~~(also sometimes just called Frechet space)~~		* [[Radial space]]
	* [[First-countable space]]		* [[First-countable space]]

Vipul: /* Symbol-free definition */

2007-12-17T19:29:09Z

Symbol-free definition

← Older revision		Revision as of 19:29, 17 December 2007
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	===Symbol-free definition===		===Symbol-free definition===

	A [[topological space]] is said to be '''sequential''' if given any subset of it which is not closed, there is a [[sequence of points]] in the subset having a [[limit]], which lies outside the subset.		A [[topological space]] is said to be '''sequential''' if given any subset of it which is not closed, there is a (possibly transfinite) [[sequence of points]] in the subset having a [[limit]], which lies outside the subset.

	==Formalisms==		==Formalisms==

Vipul at 02:48, 19 August 2007

2007-08-19T02:48:38Z

New page

{{topospace property}}

==History==

===Origin===

The notion of ''sequential space'' was introduced by S. P. Franklin in 1965.
==Definition==

===Symbol-free definition===

A [[topological space]] is said to be '''sequential''' if given any subset of it which is not closed, there is a [[sequence of points]] in the subset having a [[limit]], which lies outside the subset.

==Formalisms==

{{subspace property implication}}

A sequential space is one where:

[[sequentially closed subset]] <math>\implies</math> [[closed subset]]

Here, a sequentially closed subset is a subset that contains the limit of every convergent sequence in it.

==Relation with oher properties==

===Stronger properties===

* [[Frechet-Urysohn space]] (also sometimes just called Frechet space)
* [[First-countable space]]

==References==

* ''Spaces in which sequences suffice'' by S. P. Franklin, ''Fund. Math. 57 (1965), 107-115''
* ''Spaces in which sequences suffice II'' by S. P. Franklin, ''Fund. Math. 61 (1967), 51-56''

==External links==

* {{wp|Sequential_space}}