locally Euclidean
A locally Euclidean space $X$ is a topological space^{} that locally “looks” like ${\mathbb{R}}^{n}$. This makes it possible to talk about coordinate axes around $X$. It also gives some topological structure to the space: for example, since ${\mathbb{R}}^{n}$ is locally compact, so is $X$. However, the restriction does not induce any geometry^{} onto $X$.
Definition Suppose $X$ is a topological space. Then $X$ is called locally Euclidean if for each $x\in X$ there is a neighbourhood $U\subseteq X$, a $V\subseteq {\mathbb{R}}^{n}$, and a homeomorphism^{} $\varphi :U\to V$. Then the triple $(U,\varphi ,n)$ is called a chart for $X$.
Here, $\mathbb{R}$ is the set of real numbers, and for $n=0$ we define ${\mathbb{R}}^{0}$ as set with a single point equipped with the discrete topology.
Local dimension
Suppose $X$ is a locally Euclidean space with $x\in X$. Further, suppose $(U,\varphi ,n)$ is a chart of $X$ such that $x\in U$. Then we define the local of $X$ at $x$ is $n$. This is well defined, that is, the local dimension^{} does not depend on the chosen chart. If $({U}^{\prime},{\varphi}^{\prime},{n}^{\prime})$ is another chart with $x\in {U}^{\prime}$, then $\psi \circ {\varphi}^{1}:\varphi (U\cap {U}^{\prime})\to \psi (U\cap {U}^{\prime})$ is a homeomorphism between $\varphi (U\cap {U}^{\prime})\subseteq {\mathbb{R}}^{n}$ and $\psi (U\cap {U}^{\prime})\subseteq {\mathbb{R}}^{{n}^{\prime}}$. By Brouwer’s theorem for the invariance of dimension (which is nontrivial), it follows that $n={n}^{\prime}$.
If the local dimension is constant, say $n$, we say that the dimension of $X$ is $n$, and write $dimX=n$.
Examples

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Any set with the discrete topology, is a locally Euclidean of dimension $0.$

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Any open subset of ${\mathbb{R}}^{n}$ is locally Euclidean.

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Any manifold is locally Euclidean. For example, using a stereographic projection, one can show that the sphere ${S}^{n}$ is locally Euclidean.
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Notes
The concept locally Euclidean has a different meaning in the setting of Riemannian manifolds.
References
 1 L. Conlon, Differentiable Manifolds: A first course, Birkhäuser, 1993.
Title  locally Euclidean 

Canonical name  LocallyEuclidean 
Date of creation  20130322 14:14:49 
Last modified on  20130322 14:14:49 
Owner  matte (1858) 
Last modified by  matte (1858) 
Numerical id  14 
Author  matte (1858) 
Entry type  Definition 
Classification  msc 5300 
Related topic  Manifold 
Related topic  LocallyHomeomorphic 
Related topic  EmptyProduct 
Defines  locally Euclidean space 
Defines  chart 