In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces is said to be an étale space Local homeomorphisms are used in the study of sheaves. Typical examples of local homeomorphisms are covering maps.

is locally homeomorphic has a neighborhood that is homeomorphic For example, a manifold

but the converse is not always true. For example, the two dimensional sphere

Formal definition

between two topological spaces is called a has an open neighborhood whose image and the restriction is a homeomorphism (where the respective subspace topologies).

Examples and sufficient conditions

Covering maps

(so that geometrically, this map wraps the real line around the circle times (that is, has winding number but it is a homeomorphism only when it is bijective).

Generalizing the previous two examples, every covering map is a local homeomorphism; in particular, the universal cover is a proper local homeomorphism between two Hausdorff spaces is also locally compact is a covering map.

Inclusion maps of open subsets

is equipped with the subspace topology) then the inclusion map is always a topological embedding

Invariance of domain

Invariance of domain is a continuous injective map is a homeomorphism will be a local homeomorphism if and only if it is a locally injective map has a neighborhood is injective).

Local homeomorphisms in analysis

It is shown in complex analysis that a complex analytic is an open subset of the complex plane) is a local homeomorphism precisely when the derivative is a point of "ramification sheets come together there).

Using the inverse function theorem). An analogous condition can be formulated for maps between differentiable manifolds.

Local homeomorphisms and Hausdorffness

is a Hausdorff space is not. Consider for instance the quotient space where the equivalence relation on the disjoint union

as above.

Local homeomorphisms and fibers

is a continuous open surjection between two Hausdorff second-countable is a Baire space is a normal space. If every fiber is a discrete subspace is a local homeomorphism. If every fiber is a discrete subspace is necessarily a dense subset's fibers are discrete (see this footnote between completely metrizable second-countable spaces that has discrete with additional effort (using the inverse function theorem to be a 's domain. 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.

Properties

is a local homeomorphism if and only if it is continuous, open, and locally injective has a neighborhood is a homeomorphism if and only if it is a bijective local homeomorphism.

Every fiber is a discrete subspace of its domain

has the subspace topology

Local homeomorphisms and composition of functions

The composition is also a local homeomorphism.

is also a local homeomorphism.

Preserved properties

transfers "local" topological properties in both directions:

• is locally connected is;

• is locally path-connected is;

• is locally compact is;

• is first-countable is.

As pointed out above, the Hausdorff property is not local in this sense and need not be preserved by local homeomorphisms.

Sheaves

The local homeomorphisms with codomain stand in a natural one-to-one correspondence with the sheaves this correspondence is in fact an equivalence of categories in a natural way. All of this is explained in detail in the article on sheaves.

Generalizations and analogous concepts

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.

See also

Notes

Citations

References