Locally Euclidean space: Difference between revisions
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== Definition == | == Definition == | ||
A topological space <math>X</math> is locally Euclidean of dimension <math>n</math>, if for every point <math>x\in X</math>, there exists an open neighborhood <math>U</math> of <math>x</math>, and a homeomorphism <math display="block">\phi: U\to V</math>where <math>V\subset \mathbb{R}^n</math> is open with the standard topology on <math>\mathbb{R}^n</math>. | A topological space <math>X</math> is locally Euclidean of dimension <math>n</math>, if for every point <math>x\in X</math>, there exists an open neighborhood <math>U</math> of <math>x</math>, and a homeomorphism <math display="block">\phi: U\to V</math>where <math>V\subset \mathbb{R}^n</math> is open with the standard topology on <math>\mathbb{R}^n</math>. | ||
== See also == | |||
* [[Euclidean space]] | |||
* [[Topological manifold]] | |||
* [[Homeomorphism]] | |||
* [[Chart]] | |||
* [[Atlas]] | |||
== Terminology == | |||
{{Terminology_table| | |||
{{Terminology_table/row | locally Euclidean space | espace localement euclidien | lokal euklidischer Raum | 局部 Euclid 空间 | 局部 Euclid 空間 | 局所ユークリッド空間 }} | |||
{{Terminology_table/row | Euclidean space | espace euclidien | euklidischer Raum | Euclid 空间 | Euclid 空間 | ユークリッド空間 }} | |||
}} | |||
Revision as of 11:37, 6 April 2026
A locally Euclidean space is a topological space that resembles a Euclidean space locally. Specifically, every point in a locally Euclidean space has an open neighborhood that is homeomorphic to an open subset in 
. The concept of locally Euclidean space is a central object in the definition of topological manifold.
Definition
A topological space 
is locally Euclidean of dimension 
, if for every point 
, there exists an open neighborhood 
of 
, and a homeomorphism
where 
is open with the standard topology on .
See also
Terminology
| en | fr | de | zh | ja | |
|---|---|---|---|---|---|
| locally Euclidean space | espace localement euclidien | lokal euklidischer Raum | 局部 Euclid 空间 | 局部 Euclid 空間 | 局所ユークリッド空間 |
| Euclidean space | espace euclidien | euklidischer Raum | Euclid 空间 | Euclid 空間 | ユークリッド空間 |