# Local homeomorphism

A mapping between topological spaces such that for every point there is a neighbourhood that maps homeomorphically into under (cf. Homeomorphism). Sometimes in the definition of a local homeomorphism the requirement is included and is also assumed to be open (cf. Open mapping). Examples of local homeomorphisms are: a continuously-differentiable mapping with non-zero Jacobian on an open subset of an -dimensional Euclidean space into the -dimensional Euclidean space; a covering mapping, in particular the natural mapping of a topological group onto its quotient space with respect to a discrete subgroup. If the mapping of a Čech-complete space, in particular a locally compact Hausdorff space, onto a Tikhonov space is open and countable-to-one, that is, , , then on some open everywhere-dense set in the mapping is a local homeomorphism.

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#### References

[a1] | A.V. Arkhangel'skii, V.I. Ponomarev, "Fundamentals of general topology: problems and exercises" , Reidel (1984) (Translated from Russian) |

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Local homeomorphism.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Local_homeomorphism&oldid=12229