Uniformly based space

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.

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 $$p$$.