# Uniformly based 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

*This term is nonstandard and is being used locally within the wiki. For its use outside the wiki, please define the term when using it.*

## Definition

A topological space is said to be **uniformly based** if it has a basis of open sets for which all basis elements are homeomorphic. The abstract space to which they are all homeomorphic is termed the **basis space**. Note that the basis space is *not* uniquely determined up to homeomorphism by the original space.

## Relation with other properties

### Stronger properties

- Manifold: The basis space here is Euclidean space
- Self-based space: A uniformly based space for which we can choose a basis space homeomorphic to the whole space. Note that the basis space of any uniformly based space is self-based

## Facts

For any property of topological spaces, a uniformly based space satisfies the property *locally* if and only if its basis space satisfies the property *locally* .