Local homeomorphism


In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local structure.
If is a local homeomorphism, is said to be an étale space over Local homeomorphisms are used in the study of sheaves. Typical examples of local homeomorphisms are covering maps.
A topological space is locally homeomorphic to if every point of has a neighborhood that is homeomorphic to an open subset of.
For example, a manifold of dimension is locally homeomorphic to
If there is a local homeomorphism from to, then is locally homeomorphic to, but the converse is not always true.
For example, the two dimensional sphere, being a manifold, is locally homeomorphic to the plane but there is no local homeomorphism between them.

Formal definition

Let X and Y be topological spaces.
A function is a local homeomorphism if for every point x in X there exists an open set U containing x, such that the image is open in Y and the restriction f|U : U → f is a homeomorphism.

Examples

By definition, every homeomorphism is also a local homeomorphism.
If U is an open subset of Y equipped with the subspace topology, then the inclusion map is a local homeomorphism.
Openness is essential here: the inclusion map of a non-open subset of Y never yields a local homeomorphism.
Let be the map that wraps the real line around the circle.
This is a local homeomorphism but not a homeomorphism.
Let be the map that wraps the circle around itself n times.
This is a local homeomorphism for all non-zero n, but a homeomorphism only in the cases where it is bijective, i.e. when or -1.
Generalizing the previous two examples, every covering map is a local homeomorphism; in particular, the universal cover of a space Y is a local homeomorphism.
In certain situations the converse is true. For example: if X is Hausdorff and Y is locally compact and Hausdorff and is a proper local homeomorphism, then p is a covering map.
There are local homeomorphisms where Y is a Hausdorff space and X is not.
Consider for instance the quotient space, where the equivalence relation ~ on the disjoint union of two copies of the reals identifies every negative real of the first copy with the corresponding negative real of the second copy.
The two copies of 0 are not identified and they do not have any disjoint neighborhoods, so X is not Hausdorff. One readily checks that the natural map is a local homeomorphism.
The fiber has two elements if y ≥ 0 and one element if y < 0.
Similarly, we can construct a local homeomorphisms where X is Hausdorff and Y is not: pick the natural map from to with the same equivalence relation ~ as above.
It is shown in complex analysis that a complex analytic function is a local homeomorphism precisely when the derivative is non-zero for all.
The function on an open disk around 0 is not a local homeomorphism at 0 when n is at least 2.
In that case 0 is a point of "ramification".
Using the inverse function theorem one can show that a continuously differentiable function is a local homeomorphism if the derivative Dxf is an invertible linear map for every.

An analogous condition can be formulated for maps between differentiable manifolds.

Properties

Every local homeomorphism is a continuous and open map. A bijective local homeomorphism is therefore a homeomorphism.
A local homeomorphism transfers "local" topological properties in both directions:
As pointed out above, the Hausdorff property is not local in this sense and need not be preserved by local homeomorphisms.
If is a local homeomorphism and U is an open subset of X, then the restriction f|U is also a local homeomorphism.
If and are local homeomorphisms, then the composition is also a local homeomorphism.
If is continuous, is a local homeomorphism, and a local homeomorphism, then f is also a local homeomorphism.
The local homeomorphisms with codomain Y stand in a natural one-to-one correspondence with the sheaves of sets on Y; this correspondence is in fact an equivalence of categories.
Furthermore, every continuous map with codomain Y gives rise to a uniquely defined local homeomorphism with codomain Y 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.