Locally Euclidean space

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 ${ℝ}^n$. The concept of locally Euclidean space is a central object in the definition of topological manifold.

Definition

A topological space $X$ is locally Euclidean of dimension $n$, if for every point $x\in X$, there exists an open neighborhood $U$ of $x$, and a homeomorphism $$\varphi :U\to V$$where $V\subset {ℝ}^n$ is open with the standard topology on ${ℝ}^n$.

The pair $(U,\varphi )$ is called a chart (or coordinate chart) on $X$. A collection of such charts that covers $X$ is an atlas.

See also

• Euclidean space
• Topological manifold
• Homeomorphism
• Chart
• Atlas

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
chart	carte	Karte	坐标卡	座標卡	チャート
atlas	atlas	Atlas	图册	圖冊	アトラス