Normal space - Revision history

Vipul: /* Stronger properties */

2016-04-20T02:27:30Z

Stronger properties

← Older revision		Revision as of 02:27, 20 April 2016
Line 24:		Line 24:
	\| [[Weaker than::normal Hausdorff space]] \|\| a normal space that is also [[Hausdorff space\|Hausdorff]]. Some people include Hausdorffness as part of the definition of normal space. \|\| (obvious) \|\| a set of size more than one with the trivial topology is normal but not Hausdorff \|\| {{intermediate notions short\|normal space\|normal Hausdorff space}}		\| [[Weaker than::normal Hausdorff space]] \|\| a normal space that is also [[Hausdorff space\|Hausdorff]]. Some people include Hausdorffness as part of the definition of normal space. \|\| (obvious) \|\| a set of size more than one with the trivial topology is normal but not Hausdorff \|\| {{intermediate notions short\|normal space\|normal Hausdorff space}}
	\|-		\|-
	\| [[Weaker than::~~Compact~~ Hausdorff space]] \|\| a [[compact space]] that is also [[Hausdorff space\|Hausdorff]] \|\| [[compact Hausdorff implies normal]] \|\| [[normal not implies compact]] \|\| {{intermediate notions short\|normal space\|compact Hausdorff space}}		\| [[Weaker than::compact Hausdorff space]] \|\| a [[compact space]] that is also [[Hausdorff space\|Hausdorff]] \|\| [[compact Hausdorff implies normal]] \|\| [[normal not implies compact]] \|\| {{intermediate notions short\|normal space\|compact Hausdorff space}}
	\|-		\|-
	\| [[Weaker than::hereditarily normal space]] \|\| every subspace is a [[normal space]] under the [[subspace topology]] \|\| \|\| [[normality is not hereditary]] \|\| {{intermediate notions short\|normal space\|hereditarily normal space}}		\| [[Weaker than::hereditarily normal space]] \|\| every subspace is a [[normal space]] under the [[subspace topology]] \|\| \|\| [[normality is not hereditary]] \|\| {{intermediate notions short\|normal space\|hereditarily normal space}}

Vipul: /* Stronger properties */

2016-04-20T02:27:21Z

Stronger properties

← Older revision		Revision as of 02:27, 20 April 2016
Line 21:		Line 21:
	{\| class="sortable" border="1"		{\| class="sortable" border="1"
	! Property !! Meaning !! 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 Hausdorff space]] \|\| a normal space that is also [[Hausdorff space\|Hausdorff]]. Some people include Hausdorffness as part of the definition of normal space. \|\| (obvious) \|\| a set of size more than one with the trivial topology is normal but not Hausdorff \|\| {{intermediate notions short\|normal space\|normal Hausdorff space}}
	\|-		\|-
	\| [[Weaker than::Compact Hausdorff space]] \|\| a [[compact space]] that is also [[Hausdorff space\|Hausdorff]] \|\| [[compact Hausdorff implies normal]] \|\| [[normal not implies compact]] \|\| {{intermediate notions short\|normal space\|compact Hausdorff space}}		\| [[Weaker than::Compact Hausdorff space]] \|\| a [[compact space]] that is also [[Hausdorff space\|Hausdorff]] \|\| [[compact Hausdorff implies normal]] \|\| [[normal not implies compact]] \|\| {{intermediate notions short\|normal space\|compact Hausdorff space}}

Vipul: /* Relation with other properties */

2012-01-28T17:44:16Z

Relation with other properties

Vipul at 22:09, 27 January 2012

2012-01-27T22:09:09Z

@@ Line 4: / Line 4: @@
 ===Equivalent definitions in tabular format===
 {| class="sortable" border="1"
@@ Line 15: / Line 13: @@
 |-
 | 3 || point-finite open cover has shrinking || every [[defining ingredient::point-finite collection|point-finite]] [[open cover]] possesses a [[shrinking]]. || for any point-finite open cover <math>U_i, i \in I</math> of <math>X</math>, there exists a shrinking <math>V_i, i \in I</math>: the <math>V_i</math> form an open cover and <math>\overline{V_i} \subseteq U_i</math>.
 |}

Vipul: moved Normal-minus-Hausdorff space to Normal space

2012-01-27T18:11:07Z

moved Normal-minus-Hausdorff space to Normal space

← Older revision	Revision as of 18:11, 27 January 2012
(No difference)

Vipul: Created page with "''Please see'' Convention:Hausdorffness assumption ==Definition== ===Equivalent definitions in tabular format=== Note that under some conventions, the definition below is ..."

2011-06-18T21:05:14Z

Created page with "''Please see'' Convention:Hausdorffness assumption ==Definition== ===Equivalent definitions in tabular format=== Note that under some conventions, the definition below is ..."

New page

''Please see'' [[Convention:Hausdorffness assumption]]

==Definition==

===Equivalent definitions in tabular format===

Note that under some conventions, the definition below is taken as the definition of ''normal space''. However, under the convention followed in this wiki, the definition of [[normal space]] includes the assumption that points are closed, forcing the space to be a [[T1 space]] (and further also a [[Hausdorff space]], based on the definition). The definition given here is obtained when we relax the requirement of points being closed. Note that this definition ''includes'' the normal spaces where all points are closed, but also includes some other spaces.

{| class="sortable" border="1"
! No. !! Shorthand !! A topological space is said to be normal(-minus-Hausdorff) if ... !! A topological space <math>X</math> is said to be normal(-minus-Hausdorff) if ...
|-
| 1 || separation of disjoint closed subsets by open subsets || given any two disjoint [[closed subset]]s in the topological space, there are disjoint open sets containing them. || given any two closed subsets <math>A,B \subseteq X</math> such that <math>A \cap B = \varnothing</math>, there exist disjoint open subsets <math>U,V</math> of <math>X</math> such that <math>A \subseteq U, B \subseteq V</math>, and <math>U \cap V = \varnothing</math>.
|-
| 2 || separation of disjoint closed subsets by continuous functions || given any two disjoint closed subsets, there is a continuous function taking the value <math>0</math> at one closed set and 1 at the other. || for any two closed subsets <math>A,B \subseteq X</math>, such that <math>A \cap B = \varnothing</math>, there exists a [[continuous map]] <math>f:X \to [0,1]</math> (to the [[closed unit interval]]) such that <math>f(x) = 0 \ \forall x \in A</math> and <math>f(x) = 1 \ \forall \ x \in B</math>.
|-
| 3 || point-finite open cover has shrinking || every [[defining ingredient::point-finite collection|point-finite]] [[open cover]] possesses a [[shrinking]]. || for any point-finite open cover <math>U_i, i \in I</math> of <math>X</math>, there exists a shrinking <math>V_i, i \in I</math>: the <math>V_i</math> form an open cover and <math>\overline{V_i} \subseteq U_i</math>.
|}

← Older revision		Revision as of 17:44, 28 January 2012
Line 20:		Line 20:

	{\| class="sortable" border="1"		{\| class="sortable" border="1"
			! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
			\|-
	\| [[Weaker than::Compact Hausdorff space]] \|\| a [[compact space]] that is also [[Hausdorff space\|Hausdorff]] \|\| [[compact Hausdorff implies normal]] \|\| [[normal not implies compact]] \|\| {{intermediate notions short\|normal space\|compact Hausdorff space}}		\| [[Weaker than::Compact Hausdorff space]] \|\| a [[compact space]] that is also [[Hausdorff space\|Hausdorff]] \|\| [[compact Hausdorff implies normal]] \|\| [[normal not implies compact]] \|\| {{intermediate notions short\|normal space\|compact Hausdorff space}}
	\|-		\|-
	\| [[Weaker than::~~Hereditarily~~ normal space]] \|\| every subspace is a [[normal space]] under the [[subspace topology]] \|\| \|\| [[normality is not hereditary]] \|\| {{intermediate notions short\|normal space\|hereditarily normal space}}		\| [[Weaker than::hereditarily normal space]] \|\| every subspace is a [[normal space]] under the [[subspace topology]] \|\| \|\| [[normality is not hereditary]] \|\| {{intermediate notions short\|normal space\|hereditarily normal space}}
	\|-		\|-
	\| [[Weaker than::~~Paracompact~~ Hausdorff space]] \|\| a [[paracompact space]] that is also [[Hausdorff space\|Hausdorff]] \|\| [[paracompact Hausdorff implies normal]] \|\| [[normal not implies paracompact]] \|\| {{intermediate notions short\|normal space\|paracompact Hausdorff space}}		\| [[Weaker than::paracompact Hausdorff space]] \|\| a [[paracompact space]] that is also [[Hausdorff space\|Hausdorff]] \|\| [[paracompact Hausdorff implies normal]] \|\| [[normal not implies paracompact]] \|\| {{intermediate notions short\|normal space\|paracompact Hausdorff space}}
	\|-		\|-
	\| [[Weaker than::~~Regular~~ Lindelof space]] \|\| both a [[regular space]] and a [[Lindelof space]] \|\| [[regular Lindelof implies normal]] \|\| [[normal not implies Lindelof]] \|\| {{intermediate notions short\|normal space\|regular Lindelof space}}		\| [[Weaker than::regular Lindelof space]] \|\| both a [[regular space]] and a [[Lindelof space]] \|\| [[regular Lindelof implies normal]] \|\| [[normal not implies Lindelof]] \|\| {{intermediate notions short\|normal space\|regular Lindelof space}}
	\|-		\|-
	\| [[Weaker than::~~Perfectly~~ normal space]] \|\| every [[closed subset]] is a [[G-delta subset]] \|\| [[perfectly normal implies normal]] \|\| [[normal not implies perfectly normal]] \|\| {{intermediate notions short\|normal space\|perfectly normal space}}		\| [[Weaker than::perfectly normal space]] \|\| every [[closed subset]] is a [[G-delta subset]] \|\| [[perfectly normal implies normal]] \|\| [[normal not implies perfectly normal]] \|\| {{intermediate notions short\|normal space\|perfectly normal space}}
	\|-		\|-
	\| [[Weaker than::~~Metrizable~~ space]] \|\| can be given the structure of a [[metric space]] with the same topology \|\| [[metrizable implies normal]] \|\| [[normal not implies metrizable]] \|\| {{intermediate notions short\|normal space\|metrizable space}}		\| [[Weaker than::metrizable space]] \|\| can be given the structure of a [[metric space]] with the same topology \|\| [[metrizable implies normal]] \|\| [[normal not implies metrizable]] \|\| {{intermediate notions short\|normal space\|metrizable space}}
	\|-		\|-
	\| [[Weaker than::CW-space]] \|\| the underlying topological space of a [[CW-complex]] \|\| [[CW implies normal]] \|\| [[normal not implies CW]] \|\| {{intermediate notions short\|normal space\|CW-space}}		\| [[Weaker than::CW-space]] \|\| the underlying topological space of a [[CW-complex]] \|\| [[CW implies normal]] \|\| [[normal not implies CW]] \|\| {{intermediate notions short\|normal space\|CW-space}}
	\|-		\|-
	\| [[Weaker than::~~Linearly~~ orderable space]] \|\| obtained using the order topology for some linear ordering \|\| [[linearly orderable implies normal]] \|\| [[normal not implies linearly orderable]] \|\| {{intermediate notions short\|normal space\|linearly orderable space}}		\| [[Weaker than::linearly orderable space]] \|\| obtained using the order topology for some linear ordering \|\| [[linearly orderable implies normal]] \|\| [[normal not implies linearly orderable]] \|\| {{intermediate notions short\|normal space\|linearly orderable space}}
			\|-
			\| [[Weaker than::collectionwise normal space]] \|\| \|\| \|\| \|\|
	\|-		\|-
	\| [[Weaker than::~~Collectionwise~~ normal space]] \|\| \|\| \|\| \|\|		\| [[Weaker than::monotonically normal space]] \|\| \|\| \|\| \|\|
	\|-		\|-
	\| [[Weaker than::~~Monotonically normal~~ space]] \|\| \|\| \|\| \|\|		\| [[Weaker than::ultraconnected space]]\|\| \|\| [[ultraconnected implies normal]] \|\| \|\| {{intermediate notions short\|normal space\|ultraconnected space}}
	\|}		\|}