Regular space: Difference between revisions
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===Symbol-free definition=== | ===Symbol-free definition=== | ||
A [[topological space]] is said to be ''' | 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 | * It is a [[T1 space]] viz all points are closed | ||
Revision as of 21:25, 15 December 2007
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
Stronger properties
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.
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.