Locally Euclidean space - Revision history

InfernalAtom683 at 04:56, 10 April 2026

2026-04-10T04:56:48Z

← Older revision		Revision as of 12:56, 10 April 2026
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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>.

			The pair <math>(U,\phi)</math> is called a [[chart]] (or coordinate chart) on <math>X</math>. A collection of such charts that covers <math>X</math> is an [[atlas]].

	== See also ==		== See also ==
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	{{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 \| 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 }}		{{Terminology_table/row \| Euclidean space \| espace euclidien \| euklidischer Raum \| Euclid 空间 \| Euclid 空間 \| ユークリッド空間 }}
			{{Terminology_table/row \| chart \| carte \| Karte \| 坐标卡 \| 座標卡 \| チャート }}
			{{Terminology_table/row \| atlas \| atlas \| Atlas \| 图册 \| 圖冊 \| アトラス }}
	}}		}}

InfernalAtom683 at 03:37, 6 April 2026

2026-04-06T03:37:15Z

← Older revision		Revision as of 11:37, 6 April 2026
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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 }}
			}}

InfernalAtom683 at 04:29, 24 March 2026

2026-03-24T04:29:47Z

← Older revision		Revision as of 12:29, 24 March 2026
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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>.

InfernalAtom683: Created page with "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 $\mathbb{R}^n$ . 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..."

2026-03-24T04:29:16Z

Created page with "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 <math>\mathbb{R}^n</math>. The concept of locally Euclidean space is a central object in the definition of topological manifold. == Definition == A topological space <math>X</math> is locally Euclidean of dimension <math>n</math>, if for every..."

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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 <math>\mathbb{R}^n</math>. The concept of locally Euclidean space is a central object in the definition of [[topological manifold]].

== 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>.