Regular Hausdorff space: Difference between revisions

Latest revision as of 02:04, 28 January 2012

Definition

A topological space is termed a regular Hausdorff space or a $T_{3}$ space if it satisfies the following equivalent conditions:

It is both a regular space and a Hausdorff space.
It is both a regular space and a T1 space.
It is both a regular space and a Kolmogorov space (i.e., a $T_{0}$ space).

Note that outside of point-set topology, and in many elementary treatments, the term regular space is used to stand for regular Hausdorff space.

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

In the T family (properties of topological spaces related to separation axioms), this is called: T3

Relation with other properties

Stronger properties

Property	Meaning	Intermediate notions
metrizable space	underlying topological space of a metric space	Tychonoff space\|FULL LIST, MORE INFO
CW-space	topological space arising as the underlying space of a CW-complex	\|FULL LIST, MORE INFO
perfectly normal Hausdorff space	it is normal and every closed subset is a G-delta subset	\|FULL LIST, MORE INFO
hereditarily normal Hausdorff space	every subset is normal in the subspace topology	\|FULL LIST, MORE INFO
monotonically normal Hausdorff space	(follow link for definition)	\|FULL LIST, MORE INFO
normal Hausdorff space	T1 and disjoint closed subsets can be separated by disjoint open subsets	Tychonoff space\|FULL LIST, MORE INFO
Tychonoff space	T1 and point and closed subset not containing it can be separated by continuous function	\|FULL LIST, MORE INFO
compact Hausdorff space	compact and Hausdorff	Tychonoff space\|FULL LIST, MORE INFO
locally compact Hausdorff space	locally compact and Hausdorff	\|FULL LIST, MORE INFO
paracompact Hausdorff space		\|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
regular space				\|FULL LIST, MORE INFO
Hausdorff space				Urysohn space\|FULL LIST, MORE INFO
T1 space				Hausdorff space, Urysohn space\|FULL LIST, MORE INFO
Kolmogorov space				Hausdorff space, Urysohn space\|FULL LIST, MORE INFO

@@ Line 5: / Line 5: @@
 # It is both a [[regular space]] and a [[defining ingredient::Hausdorff space]].
 # It is both a [[regular space]] and a [[defining ingredient::T1 space]].
+# It is both a [[regular space]] and a [[defining ingredient::Kolmogorov space]] (i.e., a <math>T_0</math> space).
 Note that outside of point-set topology, and in many elementary treatments, the term ''regular space'' is used to stand for regular Hausdorff space.
@@ Line 11: / Line 12: @@
 {{T family|T3}}
+==Relation with other properties==
+===Stronger properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Weaker than::metrizable space]] || underlying topological space of a [[metric space]] ||  || || {{intermediate notions short|regular Hausdorff space|metrizable space}}
+|-
+| [[Weaker than::CW-space]] || topological space arising as the underlying space of a [[CW-complex]] || || || {{intermediate notions short|regular Hausdorff space|CW-space}}
+|-
+| [[Weaker than::perfectly normal Hausdorff space]] || it is [[normal space|normal]] and every [[closed subset]] is a [[G-delta subset]] || || || {{intermediate notions short|regular Hausdorff space|perfectly normal Hausdorff space}}
+|-
+| [[Weaker than::hereditarily normal Hausdorff space]] || every subset is normal in the subspace topology || || || {{intermediate notions short|regular Hausdorff space|hereditarily normal Hausdorff space}}
+|-
+| [[Weaker than::monotonically normal Hausdorff space]] || (follow link for definition) || || || {{intermediate notions short|regular Hausdorff space|monotonically normal Hausdorff space}}
+|-
+| [[Weaker than::normal Hausdorff space]] || T1 and disjoint closed subsets can be separated by disjoint open subsets || || || {{intermediate notions short|regular Hausdorff space|normal Hausdorff space}}
+|-
+| [[Weaker than::Tychonoff space]] || T1 and point and closed subset not containing it can be separated by continuous function || || || {{intermediate notions short|regular Hausdorff space|Tychonoff space}}
+|-
+| [[Weaker than::compact Hausdorff space]] || [[compact space|compact]] and [[Hausdorff space|Hausdorff]] || || || {{intermediate notions short|regular Hausdorff space|compact Hausdorff space}}
+|-
+| [[Weaker than::locally compact Hausdorff space]] || [[locally compact space|locally compact]] and [[Hausdorff space|Hausdorff]] || || || {{intermediate notions short|regular Hausdorff space|locally compact Hausdorff space}}
+|-
+| [[Weaker than::paracompact Hausdorff space]] || || || || {{intermediate notions short|regular Hausdorff space|paracompact Hausdorff space}}
+|}
+===Weaker properties===
+{| class="sortable" border="1"
+! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
+|-
+| [[Stronger than::regular space]] || || || || {{intermediate notions short|regular space|regular Hausdorff space}}
+|-
+| [[Stronger than::Hausdorff space]] || || || || {{intermediate notions short|Hausdorff space|regular Hausdorff space}}
+|-
+| [[Stronger than::T1 space]] || || || || {{intermediate notions short|T1 space|regular Hausdorff space}}
+|-
+| [[Stronger than::Kolmogorov space]] || || || || {{intermediate notions short|Kolmogorov space|regular Hausdorff space}}
+|}