Locally regular space

In mathematics, particularly topology, a topological space "X" is locally regular if intuitively it looks locally like a regular 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 the subspace 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 a neighbourhood that is regular under the subspace 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 compact Hausdorff space is always locally regular.
* A regular space is always locally regular
* A T1 space need not be locally regular as the set of all real numbers endowed with the cofinite 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


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