local homeomorphism

Definition. Let X and Y be topological spacesMathworldPlanetmath. Continuous mapMathworldPlanetmath f:XY is said to be locally invertible in xX iff there exist open subsets UX and VY such that xU, f(x)V and the restrictionPlanetmathPlanetmathPlanetmath


is a homeomorphism. If f is locally invertible in every point of X, then f is called a local homeomorphism.

Examples. Of course every homeomorphism is a local homeomorphism, but the converseMathworldPlanetmath is not true. For example, let f: be an exponential functionDlmfDlmfMathworld, i.e. f(z)=ez. Then f is a local homeomorphism, but it is not a homeorphism (indeed, f(z)=f(z+2πi) for any z).

One of the most important theorem of differential calculus (i.e. inverse function theoremMathworldPlanetmath) states, that if f:MN is a C1-map between C1-manifolds such that Txf:TxMTf(x)N is a linear isomorphism for a given xM, then f is locally invertible in x (in this case the local inversePlanetmathPlanetmathPlanetmathPlanetmath is even a C1-map).

Title local homeomorphism
Canonical name LocalHomeomorphism
Date of creation 2013-03-22 18:53:47
Last modified on 2013-03-22 18:53:47
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Definition
Classification msc 54C05