Completely regular space
From Topospaces
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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Definition
A topological space is termed completely regular if it satisfies the following equivalent conditions:
No. | Shorthand | A topological space is termed completely regular if ... | A topological space is termed completely regular if ... |
---|---|---|---|
1 | continuous function separating point and closed subset | given any point and any closed subset, there is a continuous map from the topological space to the closed unit interval that takes the value at the point and at the closed subset. | given any point and closed subset such that , there exists a continuous map such that and for all . |
2 | uniform structure | it occurs as the underlying topological space of a uniform space. | there is a uniform space structure on . |
Convention issues
Note that in some conventions, the assumption is made along with completely regular. We use the term Tychonoff space here for a completely regular space that is also .
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
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
regular space (also called ) | , and disjoint open subsets separating point and disjoint closed subset | completely regular implies regular | regular not implies completely regular | |FULL LIST, MORE INFO |
Urysohn space | continuous function to separating any two distinct points | completely regular implies Urysohn | Urysohn not implies completely regular | |FULL LIST, MORE INFO |
Hausdorff space (also called ) | distinct points can be separated by disjoint open subsets | (via regular) | (via regular) | |FULL LIST, MORE INFO |
T1 space | points are closed | by definition | |FULL LIST, MORE INFO |
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
subspace-hereditary property of topological spaces | Yes | complete regularity is hereditary | If is a completely regular space and is a subset of , then is completely regular with the subspace topology. |
product-closed property of topological spaces | Yes | complete regularity is product-closed | If , is a family of completely regular spaces, the product space is also a completely regular space with the product topology. |
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)