Completely regular space - Revision history

Vipul: /* Definition */

2012-01-27T18:02:15Z

Definition

← Older revision		Revision as of 18:02, 27 January 2012
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	==Definition==		==Definition==

	A [[topological space]] is termed '''completely regular''' if it ~~is a [[T1 space]] and~~ satisfies the following equivalent conditions:		A [[topological space]] is termed '''completely regular''' if it satisfies the following equivalent conditions:

	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! No. !! Shorthand !! A ~~[[T1 space\|T1]]~~ topological space is termed completely regular if ... !! A ~~[[T1 space\|T1]]~~ topological space <math>X</math> is termed completely regular if ...		! No. !! Shorthand !! A topological space is termed completely regular if ... !! A topological space <math>X</math> is termed completely regular if ...
	\|-		\|-
	\| 1 \|\| continuous function separating point and closed subset \|\| given any point and any closed subset, there is a [[continuous map]] from the topological space to the [[closed unit interval]] <math>[0,1]</math> that takes the value <math>0</math> at the point and <math>1</math> at the closed subset. \|\| given any point <math>x \in X</math> and closed subset <math>A \subseteq X</math> such that <math>x \notin A</math>, there exists a [[continuous map]]<math>f:X \to [0,1]</math> such that <math>f(x) = 0</math> and <math>f(a) = 1</math> for all <math>a \in A</math>.		\| 1 \|\| continuous function separating point and closed subset \|\| given any point and any closed subset, there is a [[continuous map]] from the topological space to the [[closed unit interval]] <math>[0,1]</math> that takes the value <math>0</math> at the point and <math>1</math> at the closed subset. \|\| given any point <math>x \in X</math> and closed subset <math>A \subseteq X</math> such that <math>x \notin A</math>, there exists a [[continuous map]]<math>f:X \to [0,1]</math> such that <math>f(x) = 0</math> and <math>f(a) = 1</math> for all <math>a \in A</math>.
	\|-		\|-
	\| 2 \|\| uniform structure \|\| it occurs as the underlying topological space of a [[uniform space]]. \|\| there is a uniform space structure <math>\mathcal{U}</math> on <math>X</math>.		\| 2 \|\| uniform structure \|\| it occurs as the underlying topological space of a [[uniform space]]. \|\| there is a uniform space structure <math>\mathcal{U}</math> on <math>X</math>.
	\|-
	\| 3 \|\| has a [[defining ingredient::compactification]] \|\| there is a [[defining ingredient::compact Hausdorff space]] having a dense subspace (with the [[subspace topology]]) [[homeomorphism\|homeomorphic]] to it. (note: T1 assumption redundant in this case) \|\| there is a [[compact Hausdorff space]] <math>C</math> and a dense subspace <math>U</math> of <math>X</math> such that <math>X</math> is homeomorphic to <math>U</math>.
	\|-
	\| 4 \|\| contained in compact Hausdorff \|\| it is homeomorphic to a subspace (not necessarily dense) of a [[compact Hausdorff space]] (note: T1 assumption redundant in this case). \|\| there is a [[compact Hausdorff space]] <math>C</math> and a subspace <math>U</math> of <math>X</math> such that <math>X</math> is homeomoephic to <math>U</math>.
	\|}		\|}

	===Convention issues===		===Convention issues===
	Note that in some conventions, the <math>T_1</math> assumption is ~~not~~ made. ~~In this case, we call~~ a ~~space~~ completely regular ~~if, given any point and any closed set not containing it, there~~ is ~~a continuous function taking the value~~ <math>0</math> ~~at the point and <math>1</math> everywhere on the closed subset~~. ~~''This latter notion is a weaker notion of completely regular''~~
			Note that in some conventions, the <math>T_1</math> assumption is made along with completely regular. We use the term [[Tychonoff space]] here for a completely regular space that is also <math>T_1</math>.

	==Formalisms==		==Formalisms==

Vipul: /* Metaproperties */

2012-01-24T23:15:33Z

Metaproperties

← Older revision		Revision as of 23:15, 24 January 2012
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	==Metaproperties==		==Metaproperties==

	{~~{subspace~~-~~closed}}~~		{\| class="sortable" border="1"
			! Metaproperty name !! Satisfied? !! Proof !! Statement with symbols
	~~Any~~ subspace of a completely regular space is completely regular.		\|-
			\| [[satisfies metaproperty::subspace-hereditary property of topological spaces]] \|\| Yes \|\| [[complete regularity is hereditary]] \|\| If <math>X</math> is a completely regular space and <math>A</math> is a subset of <math>X</math>, then <math>A</math> is completely regular with the subspace topology.
	~~{{DP~~-closed}}		\|-
			\| [[satisfies metaproperty::product-closed property of topological spaces]] \|\| Yes \|\| [[complete regularity is product-closed]] \|\| If <math>X_i, i \in I</math>, is a family of completely regular spaces, the product space <math>\prod_{i \in I} X_i</math> is also a completely regular space with the product topology.
	~~An arbitrary~~ product of completely regular spaces is completely regular.		\|}

	==References==		==References==

Vipul: /* Definition */

2012-01-24T23:11:20Z

Definition

← Older revision		Revision as of 23:11, 24 January 2012
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	\| 2 \|\| uniform structure \|\| it occurs as the underlying topological space of a [[uniform space]]. \|\| there is a uniform space structure <math>\mathcal{U}</math> on <math>X</math>.		\| 2 \|\| uniform structure \|\| it occurs as the underlying topological space of a [[uniform space]]. \|\| there is a uniform space structure <math>\mathcal{U}</math> on <math>X</math>.
	\|-		\|-
	\| 3 \|\| has a [[defining ingredient::compactification]] \|\| there is a [[defining ingredient::compact Hausdorff space]] having a dense subspace (with the [[subspace topology]]) [[homeomorphism\|homeomorphic]] to it. (note: T1 assumption redundant in this case) \|\| there is a [[compact Hausdorff space]] <math>C</math> and a dense subspace <math>U</math> of <math>X</math> such that <math>X</math> is ~~homeomoephic~~ to <math>U</math>.		\| 3 \|\| has a [[defining ingredient::compactification]] \|\| there is a [[defining ingredient::compact Hausdorff space]] having a dense subspace (with the [[subspace topology]]) [[homeomorphism\|homeomorphic]] to it. (note: T1 assumption redundant in this case) \|\| there is a [[compact Hausdorff space]] <math>C</math> and a dense subspace <math>U</math> of <math>X</math> such that <math>X</math> is homeomorphic to <math>U</math>.
	\|-		\|-
	\| 4 \|\| contained in compact Hausdorff \|\| it is homeomorphic to a subspace (not necessarily dense) of a [[compact Hausdorff space]] (note: T1 assumption redundant in this case). \|\| there is a [[compact Hausdorff space]] <math>C</math> and a subspace <math>U</math> of <math>X</math> such that <math>X</math> is homeomoephic to <math>U</math>.		\| 4 \|\| contained in compact Hausdorff \|\| it is homeomorphic to a subspace (not necessarily dense) of a [[compact Hausdorff space]] (note: T1 assumption redundant in this case). \|\| there is a [[compact Hausdorff space]] <math>C</math> and a subspace <math>U</math> of <math>X</math> such that <math>X</math> is homeomoephic to <math>U</math>.

Vipul: /* Relation with other properties */

2010-10-21T02:29:55Z

Relation with other properties

← Older revision		Revision as of 02:29, 21 October 2010
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	\| [[Weaker than::underlying space of T0 topological group]] \|\| Occurs as the underlying topological space of a [[T0 topological group]] \|\| \|\| \|\| {{intermediate notions short\|completely regular space\|underlying space of T0 topological group}}		\| [[Weaker than::underlying space of T0 topological group]] \|\| Occurs as the underlying topological space of a [[T0 topological group]] \|\| \|\| \|\| {{intermediate notions short\|completely regular space\|underlying space of T0 topological group}}
	\|-		\|-
	\| [[Weaker than::metrizable space]] \|\| ~~topological space arising~~ from a [[metric space]] \|\| (via normal) \|\| (via normal) \|\| {{intermediate notions short\|completely regular space\|metrizable space}}		\| [[Weaker than::metrizable space]] \|\| arises from a [[metric space]] via the [[metric induces topology\|metric-induced topology]] \|\| (via normal) \|\| (via normal) \|\| {{intermediate notions short\|completely regular space\|metrizable space}}
	\|-		\|-
	\| [[Weaker than::CW-space]] \|\| ~~topological space arising~~ from a [[CW-complex]] \|\| (via normal) \|\| (via normal) \|\| {{intermediate notions short\|completely regular space\|CW-space}}		\| [[Weaker than::CW-space]] \|\| arises from a [[CW-complex]] \|\| (via normal) \|\| (via normal) \|\| {{intermediate notions short\|completely regular space\|CW-space}}
	\|-		\|-
	\| [[Weaker than::polyhedron]] \|\| geometric realization of a [[simplicial complex]] \|\| (via normal) \|\| (via normal) \|\| {{intermediate notions short\|completely regular space\|polyhedron}}		\| [[Weaker than::polyhedron]] \|\| geometric realization of a [[simplicial complex]] \|\| (via normal) \|\| (via normal) \|\| {{intermediate notions short\|completely regular space\|polyhedron}}

Vipul: /* Definition */

2010-10-21T02:28:59Z

Definition

← Older revision		Revision as of 02:28, 21 October 2010
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	\| 3 \|\| has a [[defining ingredient::compactification]] \|\| there is a [[defining ingredient::compact Hausdorff space]] having a dense subspace (with the [[subspace topology]]) [[homeomorphism\|homeomorphic]] to it. (note: T1 assumption redundant in this case) \|\| there is a [[compact Hausdorff space]] <math>C</math> and a dense subspace <math>U</math> of <math>X</math> such that <math>X</math> is homeomoephic to <math>U</math>.		\| 3 \|\| has a [[defining ingredient::compactification]] \|\| there is a [[defining ingredient::compact Hausdorff space]] having a dense subspace (with the [[subspace topology]]) [[homeomorphism\|homeomorphic]] to it. (note: T1 assumption redundant in this case) \|\| there is a [[compact Hausdorff space]] <math>C</math> and a dense subspace <math>U</math> of <math>X</math> such that <math>X</math> is homeomoephic to <math>U</math>.
	\|-		\|-
	\| 4 \|\| contained in compact Hausdorff \|\| it is homeomorphic to a subspace (not necessarily dense) of a [[compact Hausdorff space]] (note: T1 assumption redundant in this case). there is a [[compact Hausdorff space]] <math>C</math> and a subspace <math>U</math> of <math>X</math> such that <math>X</math> is homeomoephic to <math>U</math>.		\| 4 \|\| contained in compact Hausdorff \|\| it is homeomorphic to a subspace (not necessarily dense) of a [[compact Hausdorff space]] (note: T1 assumption redundant in this case). \|\| there is a [[compact Hausdorff space]] <math>C</math> and a subspace <math>U</math> of <math>X</math> such that <math>X</math> is homeomoephic to <math>U</math>.
	\|}		\|}

Vipul: /* Definition */

2010-10-21T02:28:35Z

Definition

← Older revision		Revision as of 02:28, 21 October 2010
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	==Definition==		==Definition==

	A [[topological space]] is termed '''completely regular''' if it satisfies the following equivalent conditions:		A [[topological space]] is termed '''completely regular''' if it is a [[T1 space]] and satisfies the following equivalent conditions:

	~~# It~~ is [[T1 space\|T1]], and, given any point and any closed subset, there is a continuous ~~function on~~ the topological space that takes the value <math>0</math> at the point and <math>1</math> at the closed subset.		{\| class="sortable" border="1"
	~~# It is~~ [[~~defining ingredient~~:~~:T1 space\|T1]~~] and occurs as the underlying topological space of a [[uniform space]].		! No. !! Shorthand !! A [[T1 space\|T1]] topological space is termed completely regular if ... !! A [[T1 space\|T1]] topological space <math>X</math> is termed completely regular if ...
	~~# It possesses~~ a [[defining ingredient::compactification]]: there is a [[defining ingredient::compact Hausdorff space]] having a dense subspace (with the [[subspace topology]]) homeomorphic to it.		\|-
	~~# It~~ is homeomorphic to a subspace (not necessarily dense) of a [[compact Hausdorff space]].		\| 1 \|\| continuous function separating point and closed subset \|\| given any point and any closed subset, there is a [[continuous map]] from the topological space to the [[closed unit interval]] <math>[0,1]</math> that takes the value <math>0</math> at the point and <math>1</math> at the closed subset. \|\| given any point <math>x \in X</math> and closed subset <math>A \subseteq X</math> such that <math>x \notin A</math>, there exists a [[continuous map]]<math>f:X \to [0,1]</math> such that <math>f(x) = 0</math> and <math>f(a) = 1</math> for all <math>a \in A</math>.
			\|-
			\| 2 \|\| uniform structure \|\| it occurs as the underlying topological space of a [[uniform space]]. \|\| there is a uniform space structure <math>\mathcal{U}</math> on <math>X</math>.
			\|-
			\| 3 \|\| has a [[defining ingredient::compactification]] \|\| there is a [[defining ingredient::compact Hausdorff space]] having a dense subspace (with the [[subspace topology]]) [[homeomorphism\|homeomorphic]] to it. (note: T1 assumption redundant in this case) \|\| there is a [[compact Hausdorff space]] <math>C</math> and a dense subspace <math>U</math> of <math>X</math> such that <math>X</math> is homeomoephic to <math>U</math>.
			\|-
			\| 4 \|\| contained in compact Hausdorff \|\| it is homeomorphic to a subspace (not necessarily dense) of a [[compact Hausdorff space]] (note: T1 assumption redundant in this case). there is a [[compact Hausdorff space]] <math>C</math> and a subspace <math>U</math> of <math>X</math> such that <math>X</math> is homeomoephic to <math>U</math>.
			\|}

	===Convention issues===		===Convention issues===

Vipul: /* Relation with other properties */

2010-10-21T02:22:46Z

Relation with other properties

← Older revision		Revision as of 02:22, 21 October 2010
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	\| [[Weaker than::locally compact Hausdorff space]] \|\| [[Hausdorff space\|Hausdorff]] and [[locally compact space\|locally compact]] \|\| [[locally compact Hausdorff implies completely regular]] \|\| [[completely regular not implies locally compact Hausdorff]] \|\| {{intermediate notions short\|completely regular space\|locally compact Hausdorff space}}		\| [[Weaker than::locally compact Hausdorff space]] \|\| [[Hausdorff space\|Hausdorff]] and [[locally compact space\|locally compact]] \|\| [[locally compact Hausdorff implies completely regular]] \|\| [[completely regular not implies locally compact Hausdorff]] \|\| {{intermediate notions short\|completely regular space\|locally compact Hausdorff space}}
	\|-		\|-
	\| [[Weaker than::underlying space of T0 topological group]] \|\| Occurs as the underlying topological space of a [[T0 topological group]] \|\| {{intermediate notions short\|completely regular space\|underlying space of T0 topological group}}		\| [[Weaker than::underlying space of T0 topological group]] \|\| Occurs as the underlying topological space of a [[T0 topological group]] \|\| \|\| \|\| {{intermediate notions short\|completely regular space\|underlying space of T0 topological group}}
	\|-		\|-
	\| [[Weaker than::metrizable space]] \|\| topological space arising from a [[metric space]] \|\| (via normal) \|\| (via normal) \|\| {{intermediate notions short\|completely regular space\|metrizable space}}		\| [[Weaker than::metrizable space]] \|\| topological space arising from a [[metric space]] \|\| (via normal) \|\| (via normal) \|\| {{intermediate notions short\|completely regular space\|metrizable space}}
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	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! ~~property~~ !! ~~quick description~~ !! ~~proof~~ of implication !! ~~proof~~ of strictness (reverse implication failure) !! ~~intermediate~~ notions		! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Stronger than::regular space]] (also called <math>T_3</math>) \|\| <math>T_1</math>, and disjoint open subsets separating point and disjoint closed subset \|\| [[completely regular implies regular]] \|\| [[regular not implies completely regular]] \|\| {{intermediate notions short\|regular space\|completely regular space}}		\| [[Stronger than::regular space]] (also called <math>T_3</math>) \|\| <math>T_1</math>, and disjoint open subsets separating point and disjoint closed subset \|\| [[completely regular implies regular]] \|\| [[regular not implies completely regular]] \|\| {{intermediate notions short\|regular space\|completely regular space}}

Vipul: /* Relation with other properties */

2010-10-21T02:22:03Z

Relation with other properties

← Older revision		Revision as of 02:22, 21 October 2010
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	===Stronger properties===		===Stronger properties===

	{\| class="~~wikitable~~" border="1"		{\| class="sortable" border="1"
	! ~~property~~ !! ~~quick description~~ !! ~~proof~~ of implication !! ~~proof~~ of strictness (reverse implication failure) !! ~~intermediate~~ notions		! Property !! Meaning!! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
	\|-		\|-
	\| [[Weaker than::~~Normal~~ space]] \|\| disjoint closed subsets can be separated by a continuous function to <math>[0,1]</math> \|\| [[normal implies completely regular]] \|\| [[completely regular not implies normal]] \|\| {{intermediate notions short\|completely regular space\|normal space}}		\| [[Weaker than::normal space]] \|\| disjoint closed subsets can be separated by a continuous function to <math>[0,1]</math> \|\| [[normal implies completely regular]] \|\| [[completely regular not implies normal]] \|\| {{intermediate notions short\|completely regular space\|normal space}}
	\|-		\|-
	\| [[Weaker than::~~Locally~~ compact Hausdorff space]] \|\| [[Hausdorff space\|Hausdorff]] and [[locally compact space\|locally compact]] \|\| [[locally compact Hausdorff implies completely regular]] \|\| [[completely regular not implies locally compact Hausdorff]] \|\| {{intermediate notions short\|completely regular space\|locally compact Hausdorff space}}		\| [[Weaker than::locally compact Hausdorff space]] \|\| [[Hausdorff space\|Hausdorff]] and [[locally compact space\|locally compact]] \|\| [[locally compact Hausdorff implies completely regular]] \|\| [[completely regular not implies locally compact Hausdorff]] \|\| {{intermediate notions short\|completely regular space\|locally compact Hausdorff space}}
	\|-		\|-
	\| [[Weaker than::~~Underlying~~ space of T0 topological group]] \|\| Occurs as the underlying topological space of a [[T0 topological group]] \|\| {{intermediate notions short\|completely regular space\|underlying space of T0 topological group}}		\| [[Weaker than::underlying space of T0 topological group]] \|\| Occurs as the underlying topological space of a [[T0 topological group]] \|\| {{intermediate notions short\|completely regular space\|underlying space of T0 topological group}}
	\|-		\|-
	\| [[Weaker than::~~Metrizable~~ space]] \|\| topological space arising from a [[metric space]] \|\| (via normal) \|\| (via normal) \|\| {{intermediate notions short\|completely regular space\|metrizable space}}		\| [[Weaker than::metrizable space]] \|\| topological space arising from a [[metric space]] \|\| (via normal) \|\| (via normal) \|\| {{intermediate notions short\|completely regular space\|metrizable space}}
	\|-		\|-
	\| [[Weaker than::CW-space]] \|\| topological space arising from a [[CW-complex]] \|\| (via normal) \|\| (via normal) \|\| {{intermediate notions short\|completely regular space\|CW-space}}		\| [[Weaker than::CW-space]] \|\| topological space arising from a [[CW-complex]] \|\| (via normal) \|\| (via normal) \|\| {{intermediate notions short\|completely regular space\|CW-space}}
	\|-		\|-
	\| [[Weaker than::~~Polyhedron~~]] \|\| geometric realization of a [[simplicial complex]] \|\| (via normal) \|\| (via normal) \|\| {{intermediate notions short\|completely regular space\|polyhedron}}		\| [[Weaker than::polyhedron]] \|\| geometric realization of a [[simplicial complex]] \|\| (via normal) \|\| (via normal) \|\| {{intermediate notions short\|completely regular space\|polyhedron}}
	\|}		\|}

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

	{\| class="~~wikitable~~" border="1"		{\| class="sortable" border="1"
	! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions		! property !! quick description !! proof of implication !! proof of strictness (reverse implication failure) !! intermediate notions
	\|-		\|-
	\| [[Stronger than::~~Regular~~ space]] (also called <math>T_3</math>) \|\| <math>T_1</math>, and disjoint open subsets separating point and disjoint closed subset \|\| [[completely regular implies regular]] \|\| [[regular not implies completely regular]] \|\| {{intermediate notions short\|regular space\|completely regular space}}		\| [[Stronger than::regular space]] (also called <math>T_3</math>) \|\| <math>T_1</math>, and disjoint open subsets separating point and disjoint closed subset \|\| [[completely regular implies regular]] \|\| [[regular not implies completely regular]] \|\| {{intermediate notions short\|regular space\|completely regular space}}
	\|-		\|-
	\| [[Stronger than::Urysohn space]] \|\| continuous function to <math>[0,1]</math> separating any two distinct points \|\| [[completely regular implies Urysohn]] \|\| [[Urysohn not implies completely regular]] \|\| {{intermediate notions short\|Urysohn space\|completely regular space}}		\| [[Stronger than::Urysohn space]] \|\| continuous function to <math>[0,1]</math> separating any two distinct points \|\| [[completely regular implies Urysohn]] \|\| [[Urysohn not implies completely regular]] \|\| {{intermediate notions short\|Urysohn space\|completely regular space}}

Vipul at 15:21, 25 October 2009

2009-10-25T15:21:58Z

← Older revision		Revision as of 15:21, 25 October 2009
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	A [[topological space]] is termed '''completely regular''' if it satisfies the following equivalent conditions:		A [[topological space]] is termed '''completely regular''' if it satisfies the following equivalent conditions:

	* It is [[T1 space\|T1]], and, given any point and any closed subset, there is a continuous function on the topological space that takes the value <math>0</math> at the point and <math>1</math> at the closed subset		# It is [[T1 space\|T1]], and, given any point and any closed subset, there is a continuous function on the topological space that takes the value <math>0</math> at the point and <math>1</math> at the closed subset.
	* It is [[T1 space\|T1]] and occurs as the underlying topological space of a [[uniform space]]		# It is [[defining ingredient::T1 space\|T1]] and occurs as the underlying topological space of a [[uniform space]].
	* It possesses a [[compactification]]		# It possesses a [[defining ingredient::compactification]]: there is a [[defining ingredient::compact Hausdorff space]] having a dense subspace (with the [[subspace topology]]) homeomorphic to it.
			# It is homeomorphic to a subspace (not necessarily dense) of a [[compact Hausdorff space]].

	Note that in some conventions, the <math>T_1</math> assumption is not made. In this case, we call a space completely regular if, given any point and any closed set not containing it, there is a continuous function taking the value <math>0</math> at the point and <math>1</math> everywhere on the closed subset.		===Convention issues===
			Note that in some conventions, the <math>T_1</math> assumption is not made. In this case, we call a space completely regular if, given any point and any closed set not containing it, there is a continuous function taking the value <math>0</math> at the point and <math>1</math> everywhere on the closed subset. ''This latter notion is a weaker notion of completely regular''

	==Formalisms==		==Formalisms==

Vipul: /* Weaker properties */

2009-10-24T14:49:59Z

Weaker properties

← Older revision		Revision as of 14:49, 24 October 2009
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	\| [[Stronger than::Urysohn space]] \|\| continuous function to <math>[0,1]</math> separating any two distinct points \|\| [[completely regular implies Urysohn]] \|\| [[Urysohn not implies completely regular]] \|\| {{intermediate notions short\|Urysohn space\|completely regular space}}		\| [[Stronger than::Urysohn space]] \|\| continuous function to <math>[0,1]</math> separating any two distinct points \|\| [[completely regular implies Urysohn]] \|\| [[Urysohn not implies completely regular]] \|\| {{intermediate notions short\|Urysohn space\|completely regular space}}
	\|-		\|-
	\| [[Stronger than::Hausdorff space]] (also called <math>T_2</math>) \|\| distinct points can be separated by disjoint open subsets \|\| (via regular) \|\| (via regular)		\| [[Stronger than::Hausdorff space]] (also called <math>T_2</math>) \|\| distinct points can be separated by disjoint open subsets \|\| (via regular) \|\| (via regular) \|\| {{intermediate notions short\|Hausdorff space\|completely regular space}}
	\|-		\|-
	\| [[Stronger than::T1 space]] \|\| points are closed \|\| by definition \|\| \|\| {{intermediate notions short\|T1 space\|completely regular space}}		\| [[Stronger than::T1 space]] \|\| points are closed \|\| by definition \|\| \|\| {{intermediate notions short\|T1 space\|completely regular space}}