# 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 is termed **locally -Euclidean** for a nonnegative integer such that it satisfies the following equivalent conditions:

- For any point , there exists an open subset such that , and is homeomorphic to the Euclidean space .
- For any point , there exists an open subset such that , and is homeomorphic to an open subset of Euclidean space .
- For any point , and any open subset , there exists an open subset of such that , and is homeomorphic to Euclidean space >

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 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 |