A locally Euclidean space is a topological space that locally “looks” like . This makes it possible to talk about coordinate axes around . It also gives some topological structure to the space: for example, since is locally compact, so is . However, the restriction does not induce any geometry onto .
Here, is the set of real numbers, and for we define as set with a single point equipped with the discrete topology.
Suppose is a locally Euclidean space with . Further, suppose is a chart of such that . Then we define the local of at is . This is well defined, that is, the local dimension does not depend on the chosen chart. If is another chart with , then is a homeomorphism between and . 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 .
Any set with the discrete topology, is a locally Euclidean of dimension
Any open subset of is locally Euclidean.
The concept locally Euclidean has a different meaning in the setting of Riemannian manifolds.
- 1 L. Conlon, Differentiable Manifolds: A first course, Birkhäuser, 1993.
|Date of creation||2013-03-22 14:14:49|
|Last modified on||2013-03-22 14:14:49|
|Last modified by||matte (1858)|
|Defines||locally Euclidean space|