A semiregular space is a topological space

Topological space

Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...

 whose regular open sets (sets that equal the interiors of their closures) form a base

Base (topology)

In mathematics, a base B for a topological space X with topology T is a collection of open sets in T such that every open set in T can be written as a union of elements of B. We say that the base generates the topology T...

.

Every regular space

Regular space

In topology and related fields of mathematics, a topological space X is called a regular space if every non-empty closed subset C of X and a point p not contained in C admit non-overlapping open neighborhoods. Thus p and C...

 is semiregular, and every topological space may be embedded into a semiregular space.

Semiregular spaces should not be confused with locally regular space

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...

s, spaces in which there is a base of open sets that induce regular subspaces. For example, the bug-eyed line is locally regular but not semiregular.