# Uniformly based space

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 $p$ of topological spaces, a uniformly based space satisfies the property locally $p$ if and only if its basis space satisfies the property locally $p$.