locally Euclidean

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

Definition Suppose X is a topological space. Then X is called locally Euclidean if for each xX there is a neighbourhood UX, a Vn, and a homeomorphismMathworldPlanetmath ϕ:UV. Then the triple (U,ϕ,n) is called a chart for X.

Here, is the set of real numbers, and for n=0 we define 0 as set with a single point equipped with the discrete topology.

Local dimension

Suppose X is a locally Euclidean space with xX. Further, suppose (U,ϕ,n) is a chart of X such that xU. Then we define the local of X at x is n. This is well defined, that is, the local dimensionMathworldPlanetmath does not depend on the chosen chart. If (U,ϕ,n) is another chart with xU, then ψϕ-1:ϕ(UU)ψ(UU) is a homeomorphism between ϕ(UU)n and ψ(UU)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 dimX=n.


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

  • Any open subset of n is locally Euclidean.

  • Any manifold is locally Euclidean. For example, using a stereographic projection, one can show that the sphere Sn 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.
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