Completely regular space: Difference between revisions
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| [[Weaker than::locally compact Hausdorff space]] || [[Hausdorff space|Hausdorff]] and [[locally compact space|locally compact]] || [[locally compact Hausdorff implies completely regular]] || [[completely regular not implies locally compact Hausdorff]] || {{intermediate notions short|completely regular space|locally compact Hausdorff space}} | | [[Weaker than::locally compact Hausdorff space]] || [[Hausdorff space|Hausdorff]] and [[locally compact space|locally compact]] || [[locally compact Hausdorff implies completely regular]] || [[completely regular not implies locally compact Hausdorff]] || {{intermediate notions short|completely regular space|locally compact Hausdorff space}} | ||
|- | |- | ||
| [[Weaker than::underlying space of T0 topological group]] || Occurs as the underlying topological space of a [[T0 topological group]] || {{intermediate notions short|completely regular space|underlying space of T0 topological group}} | | [[Weaker than::underlying space of T0 topological group]] || Occurs as the underlying topological space of a [[T0 topological group]] || || || {{intermediate notions short|completely regular space|underlying space of T0 topological group}} | ||
|- | |- | ||
| [[Weaker than::metrizable space]] || topological space arising from a [[metric space]] || (via normal) || (via normal) || {{intermediate notions short|completely regular space|metrizable space}} | | [[Weaker than::metrizable space]] || topological space arising from a [[metric space]] || (via normal) || (via normal) || {{intermediate notions short|completely regular space|metrizable space}} | ||
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{| class="sortable" border="1" | {| class="sortable" border="1" | ||
! | ! Property !! Meaning !! Proof of implication !! Proof of strictness (reverse implication failure) !! Intermediate notions | ||
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| [[Stronger than::regular space]] (also called <math>T_3</math>) || <math>T_1</math>, and disjoint open subsets separating point and disjoint closed subset || [[completely regular implies regular]] || [[regular not implies completely regular]] || {{intermediate notions short|regular space|completely regular space}} | | [[Stronger than::regular space]] (also called <math>T_3</math>) || <math>T_1</math>, and disjoint open subsets separating point and disjoint closed subset || [[completely regular implies regular]] || [[regular not implies completely regular]] || {{intermediate notions short|regular space|completely regular space}} | ||
Revision as of 02:22, 21 October 2010
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.
VIEW: Definitions built on this | Facts about this | Survey articles about this
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 at the point and at the closed subset.
- It is T1 and occurs as the underlying topological space of a uniform space.
- It possesses a compactification: there is a compact Hausdorff space having a dense subspace (with the subspace topology) homeomorphic to it.
- It is homeomorphic to a subspace (not necessarily dense) of a compact Hausdorff space.
Convention issues
Note that in some conventions, the 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 at the point and everywhere on the closed subset. This latter notion is a weaker notion of completely regular
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
![{\displaystyle [0,1]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/738f7d23bb2d9642bab520020873cccbef49768d)
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
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. MunkresMore 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. ThorpeMore info, Page 37 (formal definition)