locally Euclidean

A locally Euclidean space $X$ is a topological space that locally “looks” like $\mathbbmss{R}^{n}$ . This makes it possible to talk about coordinate axes around $X$ . It also gives some topological structure to the space: for example, since $\mathbbmss{R}^{n}$ 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 $\phi:U\to V$ . Then the triple $(U,\phi,n)$ 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 $(U,\phi,n)$ 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 $(U^{\prime},\phi^{\prime},n^{\prime})$ is another chart with $x\in U^{\prime}$ , then $\psi\circ\phi^{-1}:\phi(U\cap U^{\prime})\to\psi(U\cap U^{\prime})$ is a homeomorphism between $\phi(U\cap U^{\prime})\subseteq\mathbb{R}^{n}$ 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 $\dim X=n$ .

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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The long line is locally Euclidean of dimension one. Note that the long line is not Hausforff. [1].

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	2013-03-22 14:14:49
Last modified on	2013-03-22 14:14:49
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	14
Author	matte (1858)
Entry type	Definition
Classification	msc 53-00
Related topic	Manifold
Related topic	LocallyHomeomorphic
Related topic	EmptyProduct
Defines	locally Euclidean space
Defines	chart