Regular space

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.
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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

• Normal space
• Completely regular space

Weaker properties

• Hausdorff space
• T1 space
• T0 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 regular space is regular. For full proof, refer: Regularity is 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

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)