Local homeomorphism

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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.



[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:
This article was adapted from an original article by B.A. Pasynkov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article