Euclidean point

Definition
A point in a topological space is termed a $$m$$-Euclidean point if it satisfies the following equivalent conditions:


 * It has a neighbourhood (open set containing it) that is homeomorphic to $$\R^m$$
 * It has a neighbourhood (open set containing it) that is homeomorphic to an open set in $$\R^m$$
 * Every neighbourhood of it contains a smaller neighbourhood homeomorphic to an open set in $$\R^m$$
 * Every neighbourhood of it contains a smaller neighbourhood homeomorphic to $$\R^m$$

A point is Euclidean if it is $$m$$-Euclidean for some $$m$$. A point cannot be $$m$$-Euclidean and $$n$$-Euclidean for $$m \ne n$$.

Stronger properties

 * Hausdorff-Euclidean point
 * Closed Euclidean point

Facts

 * In a locally Euclidean space, and more specifically in a manifold, every point is Euclidean (in fact, since locally Euclidean spaces are T1, every point is closed Euclidean).