Locally Hausdorff space - Revision history

Vipul: /* Formalisms */

2016-05-30T20:40:35Z

Formalisms

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	{{obtained by applying the\|locally operator\|Hausdorff space}}		{{obtained by applying the\|locally operator\|Hausdorff space}}

			Note that since [[Hausdorffness is hereditary]], some variants of the locally operator all collapse to the same meaning. In particular, every point being contained in an open Hausdorff subset is equivalent to having a basis of open Hausdorff subsets.

	==Metaproperties==		==Metaproperties==

Vipul at 20:39, 30 May 2016

2016-05-30T20:39:41Z

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	# <math>X</math> is a union of [[open subset]]s each of which is a [[Hausdorff space]] with the [[subspace topology]].		# <math>X</math> is a union of [[open subset]]s each of which is a [[Hausdorff space]] with the [[subspace topology]].
	# <math>X</math> has a [[basis]] comprising [[Hausdorff space]]s.		# <math>X</math> has a [[basis]] comprising [[Hausdorff space]]s.

			{{topospace property}}

			{{variation of\|Hausdorff space}}

	== Formalisms ==		== Formalisms ==
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	{{obtained by applying the\|locally operator\|Hausdorff space}}		{{obtained by applying the\|locally operator\|Hausdorff space}}

	~~{{topospace property}}~~

	~~{{variationof\|Hausdorffness}}~~

	==Metaproperties==		==Metaproperties==

Vipul: /* In terms of the local operator */

2016-05-30T20:38:42Z

In terms of the local operator

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	# <math>X</math> has a [[basis]] comprising [[Hausdorff space]]s.		# <math>X</math> has a [[basis]] comprising [[Hausdorff space]]s.

	==~~=In terms of the local operator=~~==		== Formalisms ==

	~~The notion of '''locally Hausdorff''' is~~ obtained by applying the [[locally operator~~]] to the [[~~Hausdorff space~~]] property. [[Hausdorffness is hereditary]], hence the different versions of the operator agree.~~		{{obtained by applying the\|locally operator\|Hausdorff space}}

	{{topospace property}}		{{topospace property}}

Vipul: /* In terms of the local operator */

2016-05-30T20:35:49Z

In terms of the local operator

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	===In terms of the local operator===		===In terms of the local operator===

	The notion of '''locally Hausdorff''' is obtained by applying the [[~~local~~ operator]] to the [[Hausdorff space]] property. [[Hausdorffness is hereditary]], hence the different versions of the operator agree.		The notion of '''locally Hausdorff''' is obtained by applying the [[locally operator]] to the [[Hausdorff space]] property. [[Hausdorffness is hereditary]], hence the different versions of the operator agree.

	{{topospace property}}		{{topospace property}}

Vipul: /* Definition */

2012-01-28T00:48:51Z

Definition

← Older revision		Revision as of 00:48, 28 January 2012
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	# <math>X</math> is a union of [[open subset]]s each of which is a [[Hausdorff space]] with the [[subspace topology]].		# <math>X</math> is a union of [[open subset]]s each of which is a [[Hausdorff space]] with the [[subspace topology]].
	# <math>X</math> has a [[basis]] comprising [[Hausdorff space]]s.		# <math>X</math> has a [[basis]] comprising [[Hausdorff space]]s.

			===In terms of the local operator===

			The notion of '''locally Hausdorff''' is obtained by applying the [[local operator]] to the [[Hausdorff space]] property. [[Hausdorffness is hereditary]], hence the different versions of the operator agree.

	{{topospace property}}		{{topospace property}}

Vipul at 00:43, 28 January 2012

2012-01-28T00:43:52Z

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	A [[topological space]] <math>X</math> is termed '''locally Hausdorff''' if it satisfies the following equivalent conditions:		A [[topological space]] <math>X</math> is termed '''locally Hausdorff''' if it satisfies the following equivalent conditions:

	* For every point <math>x \in X</math>, there is an [[open subset]] <math>U</math> of <math>X</math> containing <math>x</math> which is [[Hausdorff space\|Hausdorff]] in the [[subspace topology]].		# For every point <math>x \in X</math>, there is an [[open subset]] <math>U</math> of <math>X</math> containing <math>x</math> which is [[Hausdorff space\|Hausdorff]] in the [[subspace topology]].
	* For every point <math>x \in X</math>, and every open subset <math>V</math> of <math>X</matH> containing <math>x</math>, there is an open subset <math>U</math> of <math>X</math> contained in <math>V</math>, and which is [[Hausdorff space\|Hausdorff]] in the [[subspace topology]] from <math>X</math>.		# For every point <math>x \in X</math>, and every open subset <math>V</math> of <math>X</matH> containing <math>x</math>, there is an open subset <math>U</math> of <math>X</math> contained in <math>V</math>, and which is [[Hausdorff space\|Hausdorff]] in the [[subspace topology]] from <math>X</math>.
			# <math>X</math> is a union of [[open subset]]s each of which is a [[Hausdorff space]] with the [[subspace topology]].
			# <math>X</math> has a [[basis]] comprising [[Hausdorff space]]s.

	{{topospace property}}		{{topospace property}}

Vipul at 22:07, 24 January 2012

2012-01-24T22:07:14Z

← Older revision		Revision as of 22:07, 24 January 2012
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			==Definition==

			A [[topological space]] <math>X</math> is termed '''locally Hausdorff''' if it satisfies the following equivalent conditions:

			* For every point <math>x \in X</math>, there is an [[open subset]] <math>U</math> of <math>X</math> containing <math>x</math> which is [[Hausdorff space\|Hausdorff]] in the [[subspace topology]].
			* For every point <math>x \in X</math>, and every open subset <math>V</math> of <math>X</matH> containing <math>x</math>, there is an open subset <math>U</math> of <math>X</math> contained in <math>V</math>, and which is [[Hausdorff space\|Hausdorff]] in the [[subspace topology]] from <math>X</math>.

	{{topospace property}}		{{topospace property}}

	{{variationof\|Hausdorffness}}		{{variationof\|Hausdorffness}}

	==~~Definition==~~		==Metaproperties==

	~~===Symbol-free definition=~~==

	A [[topological ~~space~~]] is ~~termed '''~~locally Hausdorff~~''' if it~~ satisfies the ~~following equivalent conditions:~~		{\| class="sortable" border="1"
			!Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
	* Every point has an open ~~neighbourhood which~~ is [[~~Hausdorff space~~\|~~Hausdorff~~]]		\|-
	* Given any point, ~~and any~~ open ~~neighbourhood of the point~~, ~~there~~ is ~~a smaller open neighbourhood of the point which~~ is Hausdorff.		\| [[satisfies metaproperty::box product-closed property of topological spaces]] \|\| Yes \|\| [[local Hausdorffness is box product-closed]] \|\| If <math>X_i, i \in I</math> is a (finite or infinite) collection of locally Hausdorff topological spaces, the product of all the <math>X_i</math>s, equipped with the [[box topology]], is also locally Hausdorff.
			\|-
			\| [[satisfies metaproperty::subspace-hereditary property of topological spaces]] \|\| Yes \|\| [[local Hausdorffness is hereditary]] \|\| Suppose <math>X</math> is a locally Hausdorff space and <math>A</math> is a subset of <math>X</math>. Under the [[subspace topology]], <math>A</math> is also locally Hausdorff.
			\|-
			\| [[satisfies metaproperty::local property of topological spaces]] \|\| Yes \|\| (by definition) \|\| Suppose <math>X</math> is a locally Hausdorff space and <math>x \in X</math>. Then, there exists an open subset <math>U</math> of <math>X</math> containing <math>x</math> such that <math>U</math> is locally Hausdorff.
			\|-
			\| [[satisfies metaproperty::refining-preserved property of topological spaces]] \|\| Yes \|\| [[localHausdorffness is refining-preserved]] \|\| Suppose <math>\tau_1</math> and <math>\tau_2</math> are two topologies on a set <math>X</math>, such that <math>\tau_1 \subseteq \tau_2</math>, i.e., every subset of <math>X</math> open with respect to <math>T_1</math> is also open with respect to <math>\tau_2</math>. Then, if <math>X</math> is locally Hausdorff with respect to <math>\tau_1</math>, it is also locally Hausdorff with respect to <math>\tau_2</math>.
			\|}

	==Relation with other properties==		==Relation with other properties==
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	===Stronger properties===		===Stronger properties===

	* [[Hausdorff space]]~~: {{proofofstrictimplicationat~~\|[[Hausdorff implies locally Hausdorff]]\|[[locally Hausdorff not implies Hausdorff]]}}		{\| class="sortable" border="1"
	* [[~~Locally~~ metrizable space]]		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	* [[~~Locally~~ Euclidean space]]		\|-
			\| [[Weaker than::Hausdorff space]] \|\| \|\| [[Hausdorff implies locally Hausdorff]] \|\| [[locally Hausdorff not implies Hausdorff]] (the standard example is the [[line with two origins]]) \|\|{{intermediate notions short\|locally Hausdorff space\|Hausdorff space}}
			\|-
			\| [[Weaker than::locally metrizable space]] \|\| \|\| \|\| \|\| {{intermediate notions short\|locally Hausdorff space\|locally metrizable space}}
			\|-
			\| [[Weaker than::locally Euclidean space]] \|\| \|\| \|\| \|\|
			\|}

	===Weaker properties===		===Weaker properties===

	* [[T1 space]]		{\| class="sortable" border="1"
			! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	~~==Metaproperties==~~		\|-
			\| [[Stronger than::T1 space]] \|\| points are closed \|\| [[locally Hausdorff implies T1]] \|\| [[T1 not implies locally Hausdorff]] \|\|
	~~{{subspace-~~closed}}		\|-
			\| [[Stronger than::Kolmogorov space]] \|\| \|\| \|\| \|\|
	~~{{finite~~-~~DP-closed}}~~		\|}

	~~{{box-product-closed}}~~

	~~{{refining-preserved}}~~

	~~{{local}~~}

Vipul: 7 revisions

2008-05-11T19:48:45Z

7 revisions

← Older revision	Revision as of 19:48, 11 May 2008
(No difference)

Vipul at 19:47, 17 December 2007

2007-12-17T19:47:44Z

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	{{box-product-closed}}		{{box-product-closed}}

			{{refining-preserved}}

	{{local}}		{{local}}

Vipul: /* Metaproperties */

2007-12-17T19:44:59Z

Metaproperties

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	{{finite-DP-closed}}		{{finite-DP-closed}}

			{{box-product-closed}}

	{{local}}		{{local}}