PlanetMath: locally Euclidean

(more info)

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locally Euclidean

(Definition)

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

Definition Suppose is a topological space. Then 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 .

Here, $\mathbb{R}$ is the set of real numbers, and for we define $\mathbb{R}^0$ as set with a single point equipped with the discrete topology.

Local dimension

Suppose

is a locally Euclidean space with $x\in X$ . Further, suppose $(U,\phi, n)$ is a chart of

such that $x\in U$ . Then we define the local dimension of

. This is well defined, that is, the local dimension does not depend on the chosen chart. If $(U',\phi', n')$ is another chart with $x\in U'$ , then $\psi\circ \phi^{-1}: \phi(U\cap U') \to \psi(U\cap U')$ is a homeomorphism between $\phi(U\cap U')\subseteq \mathbb{R}^n$ and $\psi(U\cap U')\subseteq \mathbb{R}^{n'}$ . By Brouwer's theorem for the invariance of dimension (which is nontrivial), it follows that

If the local dimension is constant, say , we say that the dimension of is , and write $\dim X = n$ .

Examples

Any set with the discrete topology, is a locally Euclidean of dimension
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 is locally Euclidean.
The long line is locally Euclidean of dimension one. Note that the long line is not Hausforff. [1].