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.
Comments
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