Locally Euclidean space

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

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

Definition

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 ${\mathbb{R}}^m$.
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 ${\mathbb{R}}^m$.
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 ${\mathbb{R}}^m$>

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.

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.

Relation with other properties

Stronger properties

• Manifold: For a manifold, we assume additionally the conditions of Hausdorff and second-countable. The line with two origins is an example of a locally Euclidean space which is not a manifold, and also shows how many properties that we prove for manifolds, fail to hold for arbitrary locally Euclidean spaces.

Weaker properties

• Locally contractible space
• Locally path-connected space
• Locally metrizable space
• Locally normal space
• Locally Hausdorff space

Manifold properties not satisfied for locally Euclidean spaces

• Hausdorff space: The line with two origins is an example of a locally 1-Euclidean space that is second-countable but not Hausdorff
• Normal space: The Prufer manifold is an example of a locally 2-Euclidean space that is Hausdorff but not normal (it also fails to be second-countable)
• Metrizable space: The long line is an example of a Hausdorff, locally 1-Euclidean space that is Hausdorff and in fact normal but not metrizable (it also fails to be second-countable).
• Manifold: The dictionary plane is an example of a metrizable locally 2-Euclidean space that is not a manifold (it fails to be second-countable)