locally Euclidean

A locally Euclidean space $X$ is a topological space that locally“looks” like ${ℝ}^n$.This makes it possible to talk aboutcoordinate axes around $X$. It also gives some topological structureto the space: for example, since ${ℝ}^n$ is locally compact, so is $X$.However, the restriction does not induce any geometry onto $X$.

DefinitionSuppose $X$ is a topological space. Then $X$ iscalled locally Euclidean if for each $x\in X$ there is a neighbourhood$U\subseteq X$, a $V\subseteq {ℝ}^n$, anda 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 notdepend 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 theoremfor 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 dimensionof $X$ is $n$, and write $dimX=n$.

Examples

• Any set with the discrete topology, is a locallyEuclidean 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 thesetting of Riemannian manifolds.