View source for Locally regular space

In [[Mathematics|mathematics]], particularly [[Topology|topology]], a [[Topological space|topological space]] ''X'' is '''locally regular''' if intuitively it looks locally like a [[Regular space|regular space]]. More precisely, a locally regular space satisfies the property that each point of the space belongs to an open subset of the space that is regular under the [[Subspace topology|subspace topology]]. 

==Formal definition==

A [[Topological space|topological space]] ''X'' is said to be '''locally regular''' [[If and only if|if and only if]] each point, ''x'', of ''X'' has a neighbourhood that is [[Regular space|regular]] under the [[Subspace topology|subspace topology]]. Equivalently, a space ''X'' is locally regular if and only if the collection of all open sets that are regular under the subspace topology forms a base for the topology on ''X''.

==Examples and properties==

* Every locally regular T0 space is [[Locally Hausdorff space|locally Hausdorff]].
* A regular space is always locally regular.
* A [[Locally compact space|locally compact]] [[Hausdorff space]] is regular, hence locally regular.
* A [[T1 space]] need not be locally regular as the set of all real numbers endowed with the cofinite topology shows.

== See also ==

* [[Homeomorphism]]
* [[Locally compact space]]
* [[Locally Hausdorff space]]
* Locally metrizable space
* [[Locally normal space]]
* [[Normal space]]
* [[Semiregular space]]

== References ==

[[Category:Topology]]
[[Category:Properties of topological spaces]]


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