locally homeomorphic

Let X and Y be topological spacesMathworldPlanetmath. Then X is locally homeomorphic to Y, if for every xX there is a neighbourhood UX of x and an http://planetmath.org/node/380open set VY, such that U and V with their respective subspace topology are homeomorphic.


  • Let X={1} and Y={2,3} be discrete spaces with one resp. two elements. Since X and Y have different cardinalities, they cannot be homeomorphic. They are, however, locally homeomorphic to each other.

  • Again, let X={1} be a discrete space with one element, but now let Y={2,3} the space with topologyMathworldPlanetmath {,{2},Y}. Then X is still locally homeomorphic to Y, but Y is not locally homeomorphic to X, since the smallest neighbourhood of 3 already has more elements than X.

  • Now, let X be as in the previous examples, and Y={2,3} be http://planetmath.org/node/3120indiscrete. Then neither X is locally homeomorphic to Y nor the other way round.

  • Non-trivial examples arise with locally Euclidean spaces, especially manifolds.

Title locally homeomorphic
Canonical name LocallyHomeomorphic
Date of creation 2013-03-22 15:14:34
Last modified on 2013-03-22 15:14:34
Owner GrafZahl (9234)
Last modified by GrafZahl (9234)
Numerical id 4
Author GrafZahl (9234)
Entry type Definition
Classification msc 54-00
Synonym local homeomorphy
Related topic LocallyEuclidean