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