# 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

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

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

• Any manifold is locally Euclidean. For example, using a stereographic projection, one can show that the sphere $S^{n}$ is locally Euclidean.

• 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 LocallyEuclidean 2013-03-22 14:14:49 2013-03-22 14:14:49 matte (1858) matte (1858) 14 matte (1858) Definition msc 53-00 Manifold LocallyHomeomorphic EmptyProduct locally Euclidean space chart