In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure.
Typical examples of local homeomorphisms are covering maps.
has a neighborhood that is homeomorphic to an open subset of
For example, the two dimensional sphere, being a manifold, is locally homeomorphic to the plane
between two topological spaces is called a local homeomorphism[1] if every point
is a proper local homeomorphism between two Hausdorff spaces and if
is equipped with the subspace topology induced by
never yields a local homeomorphism (since it will not be an open map).
is a continuous injective map from an open subset
is an open subset of the complex plane
(The converse is false, as shown by the local homeomorphism
An analogous condition can be formulated for maps between differentiable manifolds.
is a continuous open surjection between two Hausdorff second-countable spaces where
-valued local homeomorphism on a dense open subset of
be the (unique) largest open subset of
's fibers are discrete (see this footnote[note 2] for an example).
between completely metrizable second-countable spaces that has discrete fibers is "almost everywhere" a local homeomorphism (in the topological sense that
is a dense open subset of its domain).
is a continuous open surjection with discrete fibers so this result guarantees that the maximal open subset
with additional effort (using the inverse function theorem for instance), it can be shown that
Because every fiber of every non-constant polynomial is finite (and thus a discrete, and even compact, subspace), this example generalizes to such polynomials whenever the mapping induced by it is an open map.
Consider for instance the quotient space
One readily checks that the natural map
In particular, every local homeomorphism is a continuous and open map.
will continue to be a local homeomorphism when it is considered as the surjective map
stand in a natural one-to-one correspondence with the sheaves of sets on
this correspondence is in fact an equivalence of categories.
gives rise to a uniquely defined local homeomorphism with codomain
The idea of a local homeomorphism can be formulated in geometric settings different from that of topological spaces.
For differentiable manifolds, we obtain the local diffeomorphisms; for schemes, we have the formally étale morphisms and the étale morphisms; and for toposes, we get the étale geometric morphisms.