# Semiregular space

From Topospaces

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

## Definition

A **semiregular space** is a topological space satisfying the following equivalent conditions:

- The regular open subsets (these are subsets that equal the interior of their closure) form a basis for the space.
- For any and any open subset containing , there exists a regular open subset of containing and contained in .

Note that *regular open* is not the same as being open and regular in the subspace topology. For the notion defined using that, see locally regular space.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

regular space | regular implies semiregular | semiregular not implies regular |