Euclidean point

This article defines a property that one can talk about for a pair of a topological space and a point in it

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

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

Relation with other properties

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).