Local homeomorphism
A mapping f : X \rightarrow Y between topological spaces such that for every point x \in X there is a neighbourhood \mathcal{O}_x that maps homeomorphically into Y under f (cf. Homeomorphism). Sometimes in the definition of a local homeomorphism the requirement f(X) = Y is included and f is also assumed to be an open mapping. Examples of local homeomorphisms are: a continuously-differentiable mapping with non-zero Jacobian on an open subset of an n-dimensional Euclidean space into the n-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 f : X \rightarrow Y of a Čech-complete space, in particular a locally compact Hausdorff space, onto a Tikhonov space Y is open and countable-to-one, that is, |f^{-1}(y)| \le \aleph_0, y \in Y, then on some open everywhere-dense set in X the mapping f 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) |
Local homeomorphism. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_homeomorphism&oldid=39586