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 an open subset of the space that is regular under the subspace topology.
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. 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.
- A regular space is always locally regular.
- A 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
- Locally Hausdorff space
- Locally compact space
- Locally metrizable space
- Normal space
- Homeomorphism
- Locally normal space
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