Locally Euclidean space: Difference between revisions

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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 $ℝ^{n}$ . 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 $ϕ : U \to V$ where $V \subset ℝ^{n}$ is open with the standard topology on $ℝ^{n}$ .

Revision as of 12:29, 24 March 2026 view source InfernalAtom683 (talk \| contribs) Bureaucrats, Interface administrators, Administrators 87 edits 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..." Tag: Visual edit		Revision as of 12:29, 24 March 2026 view source InfernalAtom683 (talk \| contribs) Bureaucrats, Interface administrators, Administrators 87 edits No edit summary Tag: Visual edit Newer edit →
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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>.