Locally Euclidean space

''Some people use the term non-Hausdorff manifold for locally Euclidean spaces that are not manifolds; however, by the convention on this wiki, Hausdorffness is part of the condition for manifolds. Learn more at convention:Hausdorffness assumption''

Locally Euclidean of a fixed dimension
A topological space $$X$$ is termed locally $$m$$-Euclidean for a nonnegative integer $$m$$ such that it satisfies the following equivalent conditions:


 * 1) For any point $$x \in X$$, there exists an open subset $$U \subseteq X$$ such that $$x \in U$$, and $$U$$ is homeomorphic to the Euclidean space $$\R^m$$.
 * 2) For any point $$x \in X$$, there exists an open subset $$U \subseteq X$$ such that $$x \in U$$, and $$U$$ is homeomorphic to an open subset of Euclidean space $$\R^m$$.
 * 3) For any point $$x \in X$$, and any open subset $$V \subseteq X$$, there exists an open subset $$U$$ of $$X$$ such that $$x \in U \subseteq V$$, and $$U$$ is homeomorphic to Euclidean space $$\R^m$$>

The equivalence of the three definitions follows from the fact that any Euclidean space is self-based: it has a basis of open subsets all of which are homeomorphic to the whole space.

Locally Euclidean of possibly varying dimension
The term locally Euclidean is also sometimes used in the case where we allow the $$m$$ to vary with the point. In other words, the equivalent conditions (1)-(3) must hold, but the nonnegative integer could vary with the point.

This case could arise if the space has multiple connected components that have different dimensions. For instance, a disjoint union of a plane and a line is locally 2-Euclidean at the points on the plane and locally Euclidean at the points on the line.

Formalisms
The equivalence between multiple sense of the term "locally" follows from the fact that any Euclidean space is self-based.