Locally Euclidean space

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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 \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 \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 \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.

Formalisms

In terms of the locally operator

This property is obtained by applying the locally operator to the property: Euclidean space

The equivalence between multiple sense of the term "locally" follows from the fact that any Euclidean space is self-based.

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
manifold We assume additionally the conditions of Hausdorff and second-countable. We also require the dimension to be the same at all points. (obvious) 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. |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
locally contractible space has a basis of contractible spaces follows from the fact that Euclidean implies contractible a pair of intersecting lines is locally contractible but not locally Euclidean (at the point of intersection) |FULL LIST, MORE INFO
locally path-connected space has a basis of path-connected spaces (via locally contractible) (via locally contractible) Locally contractible space|FULL LIST, MORE INFO
locally metrizable space every point is contained in a metrizable open subset follows from the fact that Euclidean implies metrizable, using the Euclidean metric the pair of intersecting lines is metrizable (and hence locally metrizable) but not locally Eucldiean |FULL LIST, MORE INFO
locally normal space has a basis of normal spaces (via locally metrizable) (via locally metrizable) |FULL LIST, MORE INFO
locally Hausdorff space every point is contained in a Hausdorff open subset (via locally metrizable) (via locally metrizable) |FULL LIST, MORE INFO

Manifold properties not satisfied for locally Euclidean spaces

Example Property failed Hausdorff? Second-countable? Dimension Other intermediate properties (that are always satisfied by manifold) that it satisfies
line with two origins Hausdorff space No Yes 1
Prufer manifold normal space Yes No 2
long line metrizable space Yes No 1 normal space
dictionary plane manifold Yes No 1 metrizable space, normal space