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

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

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.

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)