Regular space
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(Redirected from Regularity)
There are two alternative definitions of the term. Please see: Convention:Hausdorffness assumption
Definition
A topological space is said to be regular if it satisfies the following equivalent conditions:
No. | Shorthand | A topological space is said to be regular if ... | A topological space is said to be regular if ... |
---|---|---|---|
1 | separation of point and closed subset not containing it | given any point and a closed subset not containing it, there are disjoint open subsets containing them. | given any point and closed subset such that , there exist disjoint open subsets of such that , and . |
2 | open neighborhood contains closure of smaller open neighborhood | given any point and an open subset containing it, there is an open subset containing the point whose closure lies in the original open subset. | given any point and open subset such that , there exists an open subset such that and the closure is contained in . |
3 | separation of compact subset and closed subset | given a compact subset and a closed subset that are disjoint, there are disjoint open subsets containing them. | given any two subsets such that , is compact and is closed, there exist disjoint open subsets of such that , and . |
4 | basis element contains closure of smaller basis element | (fix a choice of basis of open subsets) given any point and a basis open subset containing it, there is a basis open subset containing the point whose closure lies in the original open subset. | given any point and a basis open subset such that , there exists a basis open subset such that and the closure is contained in . |
5 | basis element contains closure of smaller open subset | (fix a choice of basis of open subsets) given any point and a basis open subset containing it, there is an open subset containing the point whose closure lies in the original open subset. | given any point and a basis open subset such that , there exists an open subset such that and the closure is contained in . |
6 | open subset contains basis element | (fix a choice of basis of open subsets) given any point and an open subset containing it, there is a basis open subset containing the point whose closure lies in the original open subset. | given any point and an open subset such that , there exists a basis open subset such that and the closure is contained in . |
Outside of point-set topology, the term regular space is often used for a regular Hausdorff space, which is the same thing as a regular T1 space. This convention is, however, eschewed by point-set topologists.
This article defines a property of topological space that is pivotal (viz important) among currently studied properties of topological spaces
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
Relation with other properties
Conjunction with other properties
- Regular Lindelof space: Conjunction with the property of being a Lindelof space.
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
metrizable space | underlying topological space of a metric space | Completely regular space, Monotonically normal space, Moore space, Normal Hausdorff space, Regular Hausdorff space, Tychonoff space|FULL LIST, MORE INFO | ||
CW-space | topological space arising as the underlying space of a CW-complex | Completely regular space, Normal Hausdorff space, Regular Hausdorff space|FULL LIST, MORE INFO | ||
completely regular space | point and closed subset not containing it can be separated by continuous function | |FULL LIST, MORE INFO | ||
compact Hausdorff space | compact and Hausdorff | Normal Hausdorff space, Regular Hausdorff space, Tychonoff space|FULL LIST, MORE INFO | ||
locally compact Hausdorff space | locally compact and Hausdorff | Completely regular space, Regular Hausdorff space|FULL LIST, MORE INFO | ||
paracompact Hausdorff space | Normal Hausdorff space, Regular Hausdorff space|FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
semiregular space | regular implies semiregular | |||
locally regular space | ||||
preregular space | regular implies preregular | |||
symmetric space |
Metaproperties
Metaproperty name | Satisfied? | Proof | Statement with symbols |
---|---|---|---|
subspace-hereditary property of topological spaces | Yes | regularity is hereditary | If is a regular space and is a subset of , then is also a regular space under the subspace topology. |
product-closed property of topological spaces | Yes | regularity is product-closed | If is a collection of regular spaces, then the product space is also regular with the product topology. |
box product-closed property of topological spaces | Yes | regularity is box product-closed | If is a collection of regular spaces, then the product space is also regular with the box topology. |
refining-preserved property of topological spaces | No | regularity is not refining-preserved | It is possible to have a topological space that is regular but such that passing to a finer topology gives a topological space that is not regular. |
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)