Self-based space

Definition
A topological space is said to be self-based if it has a basis of open sets for which every basis set is homeomorphic to the whole space.

Stronger properties

 * Euclidean space

Weaker properties

 * Uniformly based space

Metaproperties
An open subset of a self-based space is self-based under the induced subspace topology. The idea is that we can choose a smaller basis element about each point in the subspace, that is homeomorphic to the open subset.