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

Definition

Symbol-free definition

A topological space is termed locally Euclidean if it satisfies the following equivalent properties:

• Every point has an open neighbourhood homeomorphic to Euclidean space
• Every point has an open neighbourhood homeomorphic to an open set in Euclidean space
• Given a point and an open neighbourhood of it, there is a smaller open neighbourhood contained inside that, which is homeomorphic to Euclidean space

The topological space is locally $m$-Euclidean, if all the Euclidean spaces above are ${\mathbb{R}}^m$s.

Alternatively we say that a topological space is locally Euclidean if every point in it is a Euclidean point.

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)