locally Euclidean

A locally Euclidean space $X$ is a topological space that locally “looks” like ${ℝ}^n$. This makes it possible to talk about coordinate axes around $X$. It also gives some topological structure to the space: for example, since ${ℝ}^n$ 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 $V\subseteq {ℝ}^n$, and a homeomorphism $\varphi :U\to V$. Then the triple $(U,\varphi ,n)$ is called a chart for $X$.

Here, $ℝ$ is the set of real numbers, and for $n=0$ we define ${ℝ}^0$ 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 $\varphi (U\cap U^{\prime })\subseteq {ℝ}^n$ and $\psi (U\cap U^{\prime })\subseteq {ℝ}^{n^{\prime }}$. 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

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

• Any open subset of ${ℝ}^n$ 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.

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