Monotonically normal space: Difference between revisions

Latest revision as of 22:27, 24 January 2012

Definition

Definition with symbols

A topological space $X$ is termed monotonically normal if it is a T1 space (i.e., all points are closed) and there exists an operator $G$ from ordered pairs of disjoint closed sets to open sets, such that:

For any disjoint closed subsets $A,B$ , $G(A,B)$ contains $A$ and its closure is disjoint from $B$
If $A\subseteq A'$ and $B'\subseteq B$ with all four sets being closed, $A$ disjoint from $B$ , and $A'$ disjoint from $B'$ , we have:

$G(A,B)\subseteq G(A',B')$

This is the monotonicity condition. Such an operator $G$ is termed a monotone normality operator.

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

This is a variation of normality. View other variations of normality

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
metrizable space	underlying topology of a metric space	metrizable implies monotonically normal	monotonically normal not implies metrizable	Elastic space, Protometrizable space\|FULL LIST, MORE INFO
ordered field-metrizable space	underlying topology of a space with a metric taking values in an ordered field	ordered field-metrizable implies monotonically normal	monotonically normal not implies ordered field-metrizable	\|FULL LIST, MORE INFO
linearly orderable space	order topology from a linear ordering on a set	linearly orderable implies monotonically normal	monotonically normal not implies linearly orderable	\|FULL LIST, MORE INFO
elastic space		elastic implies monotonically normal	monotonically normal not implies elastic	\|FULL LIST, MORE INFO
closed sub-Euclidean space		(via metrizable)	(via metrizable)	Elastic space, Metrizable space, Protometrizable space\|FULL LIST, MORE INFO
manifold		(via metrizable)	(via metrizable)	Elastic space, Metrizable space, Protometrizable space\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
normal space	any two disjoint closed subsets are separated by disjoint open subsets	monotonically normal implies normal	normal not implies monotonically normal	Collectionwise normal space, Hereditarily collectionwise normal space, Hereditarily normal space\|FULL LIST, MORE INFO
hereditarily normal space	every subspace is a normal space	monotonically normal implies hereditarily normal	hereditarily normal implies monotonically normal	Hereditarily collectionwise normal space\|FULL LIST, MORE INFO
collectionwise normal space	every discrete collection of closed subsets can be separated by disjoint open subsets	monotonically normal not implies collectionwise normal	collectionwise normal not implies monotonically normal	Hereditarily collectionwise normal space\|FULL LIST, MORE INFO
hereditarily collectionwise normal space	every subspace is collectionwise normal	monotonically normal implies hereditarily collectionwise normal	hereditarily collectionwise normal not implies monotonically normal	\|FULL LIST, MORE INFO
completely regular space		(via normal)	(via normal)	Normal Hausdorff space\|FULL LIST, MORE INFO
regular space		(via normal)	(via normal)	Normal Hausdorff space\|FULL LIST, MORE INFO
Hausdorff space		(via normal)	(via normal)	Normal Hausdorff space\|FULL LIST, MORE INFO
Urysohn space		(via normal)	(via normal)	\|FULL LIST, MORE INFO
collectionwise Hausdorff space		(via collectionwise normal)	(via collectionwise normal)	Collectionwise normal space, Hereditarily collectionwise normal space\|FULL LIST, MORE INFO

Incomparable properties

Perfectly normal space

Metaproperties

Hereditariness

This property of topological spaces is hereditary, or subspace-closed. In other words, any subspace (subset with the subspace topology) of a topological space with this property also has this property.
View other subspace-hereditary properties of topological spaces

Any subspace of a monotonically normal space is monotonically normal. For full proof, refer: Monotone normality is hereditary

@@ Line 1: / Line 1: @@
-{{topospace property}}
-{{variationof|normality}}
-{{source|[[wp:Monotonically normal space|Wikipedia]]}}
 ==Definition==
 ===Definition with symbols===
-A [[topological space]] <math>X</math> is termed '''monotonically normal''' if there exists an operator <math>G</math> from ordered pairs of disjoint closed sets to open sets, such that:
+A [[topological space]] <math>X</math> is termed '''monotonically normal''' if it is a [[T1 space]] (i.e., all points are closed) and there exists an operator <math>G</math> from ordered pairs of disjoint closed sets to open sets, such that:
-* For any disjoint closed subsets <math>A,B</math>, <math>G(A,B)</math> contains <math>A</math> and its closure is disjoint from <math>B</math>
+# For any disjoint closed subsets <math>A,B</math>, <math>G(A,B)</math> contains <math>A</math> and its closure is disjoint from <math>B</math>
-* If <math>A \subset A'</math> and <math>B' \subset B</math> with all four sets being closed, and <math>B</math> disjoint from <math>B'</math>, we have:
+# If <math>A \subseteq A'</math> and <math>B' \subseteq B</math> with all four sets being closed, <math>A</math> disjoint from <math>B</math>, and <math>A'</math> disjoint from <math>B'</math>, we have:
-<math>G(A,B) \subset G(A',B')</math>
+<math>G(A,B) \subseteq G(A',B')</math>
 This is the ''monotonicity'' condition. Such an operator <math>G</math> is termed a monotone normality operator.
+{{topospace property}}
+{{variationof|normality}}
 ==Relation with other properties==
@@ Line 22: / Line 20: @@
 ===Stronger properties===
-* [[Metrizable space]]
+{| class="sortable" border="1"
-* [[Ordered field-metrizable space]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Linearly orderable space]]
+|-
-* [[Elastic space]]
+| [[Weaker than::metrizable space]] || underlying topology of a [[metric space]] || [[metrizable implies monotonically normal]] || [[monotonically normal not implies metrizable]] || {{intermediate notions short|monotonically normal space|metrizable space}}
+|-
+| [[Weaker than::ordered field-metrizable space]] || underlying topology of a space with a metric taking values in an ordered field || [[ordered field-metrizable implies monotonically normal]] || [[monotonically normal not implies ordered field-metrizable]] || {{intermediate notions short|monotonically normal space|ordered field-metrizable space}}
+|-
+| [[Weaker than::linearly orderable space]] || [[order topology]] from a linear ordering on a set || [[linearly orderable implies monotonically normal]] || [[monotonically normal not implies linearly orderable]] || {{intermediate notions short|monotonically normal space|linearly orderable space}}
+|-
+| [[Weaker than::elastic space]] || || [[elastic implies monotonically normal]] || [[monotonically normal not implies elastic]] || {{intermediate notions short|monotonically normal space|elastic space}}
+|-
+| [[Weaker than::closed sub-Euclidean space]] || || (via metrizable) || (via metrizable) || {{intermediate notions short|monotonically normal space|closed sub-Euclidean space}}
+|-
+| [[Weaker than::manifold]] || || (via metrizable) || (via metrizable) || {{intermediate notions short|monotonically normal space|manifold}}
+|}
 ===Weaker properties===
-* [[Hereditarily collectionwise normal space]]
+{| class="sortable" border="1"
-* [[Hereditarily normal space]]
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
-* [[Collectionwise normal space]]
+|-
-* [[Normal space]]
+| [[Stronger than::normal space]] || any two disjoint closed subsets are separated by disjoint open subsets || [[monotonically normal implies normal]] || [[normal not implies monotonically normal]] || {{intermediate notions short|normal space|monotonically normal space}}
+|-
+| [[Stronger than::hereditarily normal space]] || every subspace is a [[normal space]] || [[monotonically normal implies hereditarily normal]] || [[hereditarily normal implies monotonically normal]] || {{intermediate notions short|hereditarily normal space|monotonically normal space}}
+|-
+| [[Stronger than::collectionwise normal space]] || every discrete collection of closed subsets can be separated by disjoint open subsets || [[monotonically normal not implies collectionwise normal]] || [[collectionwise normal not implies monotonically normal]] || {{intermediate notions short|collectionwise normal space|monotonically normal space}}
+|-
+| [[Stronger than::hereditarily collectionwise normal space]] || every subspace is [[collectionwise normal space|collectionwise normal]] || [[monotonically normal implies hereditarily collectionwise normal]] || [[hereditarily collectionwise normal not implies monotonically normal]] || {{intermediate notions short|hereditarily collectionwise normal space|monotonically normal space}}
+|-
+| [[Stronger than::completely regular space]] || || (via normal) || (via normal) || {{intermediate notions short|completely regular space|monotonically normal space}}
+|-
+| [[Stronger than::regular space]] || || (via normal) || (via normal) || {{intermediate notions short|regular space|monotonically normal space}}
+|-
+| [[Stronger than::Hausdorff space]]|| || (via normal) || (via normal) || {{intermediate notions short|Hausdorff space|monotonically normal space}}
+|-
+| [[Stronger than::Urysohn space]] || || (via normal) || (via normal) || {{intermediate notions short|Urysohn space|monotonically normal space}}
+|-
+| [[Stronger than::collectionwise Hausdorff space]] || || (via collectionwise normal) || (via collectionwise normal) || {{intermediate notions short|collectionwise Hausdorff space|monotonically normal space}}
+|}
 ===Incomparable properties===