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locally Euclidean
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(Definition)
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A locally Euclidean space $X$ is a topological space that locally ``looks'' like
 . This makes it possible to talk about coordinate axes around $X$ . It also gives some topological structure to the space: for example, since
 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 $V\subseteq \sR^n$ , and a homeomorphism $\phi: U\to V$ . Then the triple $(U,\phi, n)$ is called a chart for $X$ .
Here, $\sR$ is the set of real numbers, and for $n=0$ we define $\sR^0$ as set with a single point equipped with the discrete topology.
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 dimension of $X$ at $x$ is $n$ . 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 $\phi(U\cap U')\subseteq \sR^n$ and
$\psi(U\cap U')\subseteq \sR^{n'}$ . By Brouwer's theorem for the invariance of dimension (which is nontrivial), it follows that $n=n'$ .
If the local dimension is constant, say $n$ , we say that the dimension of $X$ is $n$ , and write $\dim X = n$ .
- Any set with the discrete topology, is a locally Euclidean of dimension $0.$
- Any open subset of $\sR^n$ 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].
The concept locally Euclidean has a different meaning in the setting of Riemannian manifolds.
- 1
- L. Conlon, Differentiable Manifolds: A first course, Birkhäuser, 1993.
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"locally Euclidean" is owned by matte. [ full author list (5) ]
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Cross-references: Riemannian manifolds, long line, sphere, stereographic projection, manifold, open subset, invariance of dimension, theorem, dimension, well defined, discrete topology, point, real numbers, homeomorphism, neighbourhood, onto, geometry, induce, restriction, locally compact, structure, coordinate, topological space
There are 14 references to this entry.
This is version 11 of locally Euclidean, born on 2004-03-12, modified 2007-03-19.
Object id is 5692, canonical name is LocallyEuclidean.
Accessed 7639 times total.
Classification:
| AMS MSC: | 53-00 (Differential geometry :: General reference works ) |
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Pending Errata and Addenda
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