Regular space: Difference between revisions
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==Metaproperties== | ==Metaproperties== | ||
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!Metaproperty name !! Satisfied? !! Proof !! Section in this article | |||
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| [[satisfies metaproperty::subspace-hereditary property of topological spaces]] || Yes || [[regularity is hereditary]] || [[#Hereditariness]] | |||
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| [[satisfies metaproperty::product-closed property of topological spaces]] || Yes || [[regularity is product-closed]] || [[#Products]] | |||
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| [[satisfies metaproperty::box product-closed property of topological spaces]]|| Yes || [[regularity is box product-closed]] || [[#Box products]] | |||
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| [[dissatisfies metaproperty::refining-preserved property of topological spaces]] || No || [[regularity is not refining-preserved]] || [[#Refining]] | |||
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{{subspace-closed}} | {{subspace-closed}} | ||
Any subspace of a regular space is regular. {{proofat|[[Regularity is hereditary]]}} | Any subspace of a regular space is regular. {{proofat|[[Regularity is hereditary]]}} | ||
{{further|[[Hausdorffness is hereditary]], [[complete regularity is hereditary]], [[normality is not hereditary]]}} | |||
{{DP-closed}} | {{DP-closed}} | ||
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An arbitrary product of regular spaces is regular. {{proofat|[[Regularity is product-closed]]}} | An arbitrary product of regular spaces is regular. {{proofat|[[Regularity is product-closed]]}} | ||
{{box | {{box product-closed}} | ||
An arbitrary box product of regular spaces is regular. {{proofat|[[Regularity is box-product-closed]]}} | An arbitrary box product of regular spaces is regular. {{proofat|[[Regularity is box-product-closed]]}} | ||
{{not refining-preserved}} | |||
Moving to a [[finer topology]] does ''not'' preserve regularity. In other words, if <math>(X,\tau)</math> is a regular space, and <math>\tau'</math> is a finer topology on <math>X</math> than <math>\tau</math>, then <math>(X,\tau')</math> need not be a regular space. {{proofat|[[Regularity is not refining-preserved]]}} | Moving to a [[finer topology]] does ''not'' preserve regularity. In other words, if <math>(X,\tau)</math> is a regular space, and <math>\tau'</math> is a finer topology on <math>X</math> than <math>\tau</math>, then <math>(X,\tau')</math> need not be a regular space. {{proofat|[[Regularity is not refining-preserved]]}} | ||
Revision as of 17:03, 26 October 2009
This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces
In the T family (properties of topological spaces related to separation axioms), this is called: T3
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
Definition
Symbol-free definition
A topological space is said to be regular if it satisfies the following two conditions:
- It is a T1 space viz all points are closed
- Given a point and a closed set not containing it, there are disjoint open sets containing the point and the closed set respectively.
Relation with other properties
Conjunction with other properties
- Regular Lindelof space: Conjunction with the property of being a Lindelof space.
Stronger properties
- Metrizable space
- CW-space
- Perfectly normal space
- Hereditarily normal space
- Monotonically normal space
- Normal space
- Completely regular space
Weaker properties
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 regular space is regular. For full proof, refer: Regularity is hereditary
Further information: Hausdorffness is hereditary, complete regularity is hereditary, normality is not hereditary
Products
This property of topological spaces is closed under taking arbitrary products
View all properties of topological spaces closed under products
An arbitrary product of regular spaces is regular. For full proof, refer: Regularity is product-closed
Box products
This property of topological spaces is a box product-closed property of topological spaces: it is closed under taking arbitrary box products
View other box product-closed properties of topological spaces
An arbitrary box product of regular spaces is regular. For full proof, refer: Regularity is box-product-closed
Refining
NO: This property of topological spaces is not a refining-preserved property of topological spaces. In other words, putting a finer topology on a topological space satisfying this property might give a topological space not satisfying this property.
View other properties that are not refining-preserved
Moving to a finer topology does not preserve regularity. In other words, if is a regular space, and is a finer topology on than , then need not be a regular space. For full proof, refer: Regularity is not refining-preserved
References
Textbook references
- Topology (2nd edition) by James R. MunkresMore 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. ThorpeMore info, Page 28 (formal definition)