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

Formal definition

A function between two topological spaces is called a if every point has an open neighborhood whose image is open in and the restriction is a homeomorphism.

Examples and sufficient conditions

Covering maps
Every homeomorphism is a local homeomorphism. The function defined by is a local homeomorphism but not a homeomorphism. The map defined by where is a fixed integer, wraps the circle around itself times and is a local homeomorphism for all non-zero 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 of a space is a local homeomorphism.
In certain situations the converse is true. For example: if is a proper local homeomorphism between two Hausdorff spaces and if is also locally compact, then is a covering map.
Inclusion maps of open subsets
If is any subspace then the inclusion map is always a topological embedding. It is a local homeomorphism if and only if is open in
Invariance of domain
Invariance of domain guarantees that if is a continuous injective map from an open subset of then is open in and is a homeomorphism.
Consequently, a continuous map from an open subset will be a local homeomorphism if and only if it is a locally injective map.
Local homeomorphisms in analysis
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, with fixed integer, defined on an open disk around, is not a local homeomorphism when
In that case 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 is an invertible linear map for every
.
An analogous condition can be formulated for maps between differentiable manifolds.
Local homeomorphisms and Hausdorffness
There exist local homeomorphisms where is a Hausdorff space but 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 are not identified and they do not have any disjoint neighborhoods, so is not Hausdorff. One readily checks that the natural map is a local homeomorphism.
The fiber has two elements if and one element if
Similarly, it is possible to construct a local homeomorphisms where is Hausdorff and is not: pick the natural map from to with the same equivalence relation as above.
Local homeomorphisms and fibers
Suppose is a continuous open surjection between two Hausdorff second-countable spaces where is a Baire space and is a normal space. If every fiber of is a discrete subspace of then is a -valued local homeomorphism on a dense open subset of
To clarify this statement's conclusion, let be the largest open subset of such that is a local homeomorphism.
If every fiber of is a discrete subspace of then this open set is necessarily a subset of
In particular, if then a conclusion that may be false without the assumption that 's fibers are discrete.
One corollary is that every continuous open surjection between completely metrizable second-countable spaces that has discrete fibers is "almost everywhere" a local homeomorphism.
For example, the map defined by the polynomial is a continuous open surjection with discrete fibers so this result guarantees that the maximal open subset is dense in with additional effort, it can be shown that which confirms that this set is indeed dense in This example also shows that it is possible for to be a dense subset of 's domain.
Because every fiber of every non-constant polynomial is finite, this example generalizes to such polynomials whenever the mapping induced by it is an open map.

Properties

A map is a local homeomorphism if and only if it is continuous, open, and locally injective. It follows that the map is a homeomorphism if and only if it is a bijective local homeomorphism.
Every fiber of a local homeomorphism is a discrete subspace of its domain
Whether or not a function is a local homeomorphism depends on its codomain: A map is a local homeomorphism if and only if the surjection is a local homeomorphism and is an open subset of

Local homeomorphisms and composition of functions

The composition of two local homeomorphisms is a local homeomorphism; explicitly, if and are local homeomorphisms then the composition is also a local homeomorphism.
The restriction of a local homeomorphism to any open subset of the domain will again be a local homeomorphism; explicitly, if is a local homeomorphism then its restriction to any open subset of is also a local homeomorphism.
If is continuous while both and are local homeomorphisms, then is also a local homeomorphism.

Preserved properties

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.

Sheaves

The local homeomorphisms with codomain stand in a natural one-to-one correspondence with the sheaves of sets on this correspondence is in fact an equivalence of categories. Furthermore, every continuous map with codomain gives rise to a uniquely defined local homeomorphism with codomain 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.