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".