Regular space: Difference between revisions

Revision as of 17:55, 27 January 2012

There are two alternative definitions of the term. Please see: Convention:Hausdorffness assumption

Definition

A topological space is said to be regular if it satisfies the following equivalent conditions:

No.	Shorthand	A topological space is said to be regular if ...	A topological space $X$ is said to be regular if ...
1	separation of point and closed subset not containing it	given any point and a closed subset not containing it, there are disjoint open subsets containing them.	given any point $x \in X$ and closed subset $A \subseteq X$ such that $x \notin A$ , there exist disjoint open subsets $U, V$ of $X$ such that $A \subseteq U, x \in V$ , and $U \cap V = \emptyset$ .
2	separation of compact subset and closed subset	given a compact subset and a closed subset that are disjoint, there are disjoint open subsets containing them.	given any two subsets $A, B \subseteq X$ such that $A \cap B = \emptyset$ , $A$ is compact and $B$ is closed, there exist disjoint open subsets $U, V$ of $X$ such that $A \subseteq U, B \subseteq V$ , and $U \cap V = \emptyset$ .

Outside of point-set topology, the term regular space is often used for a regular Hausdorff space, which is the same thing as a regular T1 space. This convention is, however, eschewed by point-set topologists.

This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces

This article is about a basic definition in topology.
VIEW: Definitions built on this | Facts about this | Survey articles about this
View a complete list of basic definitions in topology

Relation with other properties

Conjunction with other properties

Regular Lindelof space: Conjunction with the property of being a Lindelof space.

Stronger properties

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

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
semiregular space
locally regular space

Metaproperties

Metaproperty name	Satisfied?	Proof	Statement with symbols
subspace-hereditary property of topological spaces	Yes	regularity is hereditary	If $X$ is a regular space and $A$ is a subset of $X$ , then $A$ is also a regular space under the subspace topology.
product-closed property of topological spaces	Yes	regularity is product-closed	If $X_{i}, i \in I$ is a collection of regular spaces, then the product space $\prod_{i \in I} X_{i}$ is also regular with the product topology.
box product-closed property of topological spaces	Yes	regularity is box product-closed	If $X_{i}, i \in I$ is a collection of regular spaces, then the product space $\prod_{i \in I} X_{i}$ is also regular with the box topology.
refining-preserved property of topological spaces	No	regularity is not refining-preserved	It is possible to have a topological space that is regular but such that passing to a finer topology gives a topological space that is not regular.

References

Textbook references

Topology (2nd edition) by James R. Munkres^{More info}, Page 195, Chapter 4, Section 31 (formal definition, along with normal space)
Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. Thorpe^{More info}, Page 28 (formal definition)

@@ Line 3: / Line 3: @@
 ==Definition==
-A [[topological space]] is said to be '''regular''' or <math>T_3</math> if it satisfies the following equivalent conditions:
+A [[topological space]] is said to be '''regular''' if it satisfies the following equivalent conditions:
 {| class="sortable" border="1"
 ! No. !! Shorthand !! A topological space is said to be regular if ... !! A topological space <math>X</math> is said to be regular if ...
 |-
-| 1 || separation of point and closed subset not containing it || all points in it are closed sets, ''and'' given any point and a [[closed subset]] not containing it, there are disjoint open subsets containing them. || for all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, ''and'' given any point <math>x \in X</math> and closed subset <math>A \subseteq X</math> such that <math>x \notin A</math>, there exist disjoint open subsets <math>U,V</math> of <math>X</math> such that <math>A \subseteq U, x \in V</math>, and <math>U \cap V = \varnothing</math>.
+| 1 || separation of point and closed subset not containing it || given any point and a [[closed subset]] not containing it, there are disjoint open subsets containing them. || given any point <math>x \in X</math> and closed subset <math>A \subseteq X</math> such that <math>x \notin A</math>, there exist disjoint open subsets <math>U,V</math> of <math>X</math> such that <math>A \subseteq U, x \in V</math>, and <math>U \cap V = \varnothing</math>.
 |-
-| 2 || separation of compact subset and closed subset || all points in it are closed sets, ''and'' given a compact subset and a closed subset that are disjoint, there are disjoint open subsets containing them. || for all <math>x \in X</math>, the set <math>\{ x \}</math> is closed in <math>X</math>, ''and'' given any two subsets <math>A,B \subseteq X</math> such that <math>A \cap B = \varnothing</math>, <math>A</math> is compact and <math>B</math> is closed, 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 compact subset and closed subset || given a compact subset and a closed subset that are disjoint, there are disjoint open subsets containing them. || given any two subsets <math>A,B \subseteq X</math> such that <math>A \cap B = \varnothing</math>, <math>A</math> is compact and <math>B</math> is closed, 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>.
 |}
-The term ''regular'' is sometimes used without the [[T1 space]] assumption. This gives a different, weaker notion of regularity, which is referred to here as [[regular-minus-Hausdorff space]].
+Outside of point-set topology, the term ''regular space'' is often used for a [[regular Hausdorff space]], which is the same thing as a regular [[T1 space]]. This convention is, however, eschewed by point-set topologists.
 {{pivotal topospace property}}
-{{T family|T3}}
 {{basicdef}}
@@ Line 56: / Line 55: @@
 ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions
 |-
-| [[Stronger than::Hausdorff space]] || any two distinct points can be separated by disjoint open subsets || || || {{intermediate notions short|Hausdorff space|regular space}}
+| [[Stronger than::semiregular space]] || || || ||
-|-
-| [[Stronger than::T1 space]] || points are closed || || || {{intermediate notions short|T1 space|regular space}}
 |-
-| [[Stronger than::Kolmogorov space]] || || || || {{intermediate notions short|Kolmogorov space|regular space}}
+| [[Stronger than::locally regular space]] || || || ||
 |}