Completely regular space: Difference between revisions

Revision as of 20:52, 24 November 2008

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: T3.5

This article is about a basic definition in topology.
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View a complete list of basic definitions in topology

Definition

A topological space is termed completely regular if it satisfies the following equivalent conditions:

It is T1, and, given any point and any closed subset, there is a continuous function on the topological space that takes the value $0$ at the point and $1$ at the closed subset
It is T1 and occurs as the underlying topological space of a uniform space
It possesses a compactification

Note that in some conventions, the $T_{1}$ assumption is not made. In this case, we call a space completely regular if, given any point and any closed set not containing it, there is a continuous function taking the value $0$ at the point and $1$ everywhere on the closed subset.

Formalisms

In terms of the subspace operator

This property is obtained by applying the subspace operator to the property: compact Hausdorff space

Relation with other properties

Stronger properties

Weaker properties

Urysohn 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 completely regular space is completely 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 completely regular spaces is completely regular.

References

Textbook references

Topology (2nd edition) by James R. Munkres^{More info}, Page 211, Chapter 4, Section 33 (formal definition)
Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I. M. Singer and J. A. Thorpe^{More info}, Page 37 (formal definition)

@@ Line 9: / Line 9: @@
 A [[topological space]] is termed '''completely regular''' if it satisfies the following equivalent conditions:
-* Given any point and any closed subset, there is a continuous function on the topological space that takes the value <math>0</math> at the point and <math>1</math> at the closed subset
+* It is [[T1 space|T1]], and, given any point and any closed subset, there is a continuous function on the topological space that takes the value <math>0</math> at the point and <math>1</math> at the closed subset
-* It occurs as the underlying topological space of a [[uniform space]]
+* It is [[T1 space|T1]] and occurs as the underlying topological space of a [[uniform space]]
 * It possesses a [[compactification]]
+Note that in some conventions, the <math>T_1</math> assumption is not made. In this case, we call a space completely regular if, given any point and any closed set not containing it, there is a continuous function taking the value <math>0</math> at the point and <math>1</math> everywhere on the closed subset.
 ==Formalisms==