T1 space - Revision history

Vipul: /* Definition */

2012-01-26T19:21:30Z

Definition

← Older revision		Revision as of 19:21, 26 January 2012
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	===Symbol-free definition===		===Symbol-free definition===

	A [[topological space]] is termed a <math>T_1</math>-space if it satisfies the following equivalent conditions:		A [[topological space]] is termed a <math>T_1</math>-space (or '''Frechet space''' or '''accessible space''') if it satisfies the following equivalent conditions:

	# Given an ordered pair of distinct points, there is an [[open subset]] of the topological space containing the first point but not the second		# Given an ordered pair of distinct points, there is an [[open subset]] of the topological space containing the first point but not the second
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	===Definition with symbols===		===Definition with symbols===

	A [[topological space]] <math>X</math> is termed a <math>T_1</math>-space if it satisfies the following equivalent conditions:		A [[topological space]] <math>X</math> is termed a <math>T_1</math>-space (or '''Frechet space''' or '''accessible space''') if it satisfies the following equivalent conditions:

	# Given two distinct points <math>x,y \in X</math>, there exists an open subset <math>U</math> of <math>X</math> such that <math>x \in U</math> and <math>y \notin U</math>		# Given two distinct points <math>x,y \in X</math>, there exists an open subset <math>U</math> of <math>X</math> such that <math>x \in U</math> and <math>y \notin U</math>

Vipul at 15:14, 21 July 2008

2008-07-21T15:14:59Z

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 ==Definition==
 A [[topological space]] is termed a <math>T_1</math>-space if it satisfies the following equivalent conditions:
-* Given an ordered pair of distinct points, there is an [[open subset]] of the topological space containing the first point but not the second
+# Given an ordered pair of distinct points, there is an [[open subset]] of the topological space containing the first point but not the second
-* Every singleton subset is a [[closed subset]] (more loosely, all points are closed)
+# Every singleton subset is a [[closed subset]] (more loosely, all points are closed)
 ==Relation with other properties==

Vipul: 4 revisions

2008-05-11T19:59:28Z

4 revisions

← Older revision	Revision as of 19:59, 11 May 2008
(No difference)

Vipul: /* Metaproperties */

2008-04-21T21:29:14Z

Metaproperties

← Older revision		Revision as of 21:29, 21 April 2008
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	{{refining-preserved}}		{{refining-preserved}}

	If we take a <math>T_1</math>-space, and switch to a [finer topology]], the new space is also <math>T_1</math>. This is because the addition of more open sets does not disturb the fact that points are closed.		If we take a <math>T_1</math>-space, and switch to a [[finer topology]], the new space is also <math>T_1</math>. This is because the addition of more open sets does not disturb the fact that points are closed.

	{{local}}		{{local}}

Vipul: /* Metaproperties */

2008-04-21T21:29:02Z

Metaproperties

← Older revision		Revision as of 21:29, 21 April 2008
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	{{local}}		{{local}}

			==References==
			===Textbook references===
			* {{booklink\|Munkres}}, Page 99 (definition in paragraph)
			* {{booklink\|SingerThorpe}}, Page 26 (formal definition, as part of a list of definitions of separation axioms)

Vipul at 19:37, 17 December 2007

2007-12-17T19:37:37Z

← Older revision		Revision as of 19:37, 17 December 2007
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	If we take a <math>T_1</math>-space, and switch to a [finer topology]], the new space is also <math>T_1</math>. This is because the addition of more open sets does not disturb the fact that points are closed.		If we take a <math>T_1</math>-space, and switch to a [finer topology]], the new space is also <math>T_1</math>. This is because the addition of more open sets does not disturb the fact that points are closed.

			{{local}}

Vipul at 09:54, 20 August 2007

2007-08-20T09:54:40Z

New page

{{topospace property}}

{{T family|T1}}

{{basicdef}}

==Definition==

A [[topological space]] is termed a <math>T_1</math>-space if it satisfies the following equivalent conditions:

* Given an ordered pair of distinct points, there is an [[open subset]] of the topological space containing the first point but not the second
* Every singleton subset is a [[closed subset]] (more loosely, all points are closed)

==Relation with other properties==

===Stronger properties===

* [[Hausdorff space]]
* [[US-space]]

===Weaker properties===

* [[Symmetric space]]
* [[T0 space]]

==Metaproperties==

{{subspace-closed}}

Any subset of a <math>T_1</math>-space, is a <math>T_1</math>-space under the [[subspace topology]].

{{DP-closed}}

A direct product of <math>T_1</math>-spaces is <math>T_1</math>.

{{refining-preserved}}

If we take a <math>T_1</math>-space, and switch to a [finer topology]], the new space is also <math>T_1</math>. This is because the addition of more open sets does not disturb the fact that points are closed.