# Locally regular space

In

mathematics , particularlytopology , atopological space "X" is**locally regular**if intuitively it looks locally like aregular space . More precisely, a locally regular space satisfies the property that each point of the space belongs to a subset of the space that is regular under thesubspace topology . A locally regular space is somewhere between a locally Hausdorff space and a locally compact Hausdorff space.**Formal definition**A

topological space "X" is said to be**locally regular**if and only if each point, "x", of "X" has aneighbourhood that is regular under thesubspace topology .Note that not every neighbourhood of "x" has to be regular, but at least one neighbourhood of "x" has to be regular (under the subspace topology).

If a space were called locally regular

if and only if each point of the space belonged to a subset of the space that was regular under the subspace topology, then every topological space would be locally regular. This is because, the singleton {"x"} is vacuously regular and contains "x". Therefore, the definition is more restricitive.**Examples and properties*** Every locally regular space is locally Hausdorff

* A locally compactHausdorff space is always locally regular.

* A regular space is always locally regular

* AT1 space need not be locally regular as the set of all real numbers endowed with thecofinite topology shows.

* A 0-dimensional space is a topological space that has a base consisting entirely of open and closed sets. Every 0-dimensional locally regular space is regular**Theorems****Theorem 1**If "X" is homeomorphic to "Y" and "X" is locally regular, then so is "Y".

**Proof**This follows from the fact that the image of a regular space under a homeomorphism is always regular.

**ee also***

Locally Hausdorff space

*Locally compact space

*Locally metrizable space

*Normal space

*Homeomorphism

*Locally normal space **References**

*Wikimedia Foundation.
2010.*

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