In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) 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.
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 (in either direction).
Let and be topological spaces. A function is a local homeomorphism if for every point there exists an open set containing such that the image is open in and the restriction is a homeomorphism (where the respective subspace topologies are used on and on ).
Examples and sufficient conditions
Every homeomorphism is also a local homeomorphism. If is a local homeomorphism and is an open subset of then the restriction is also a local homeomorphism. The composition of two local homeomorphisms is a local homeomorphism (that is, 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 is also a local homeomorphism.
If is an open subset of equipped with the subspace topology, then the inclusion map is a local homeomorphism. The subset being open in is essential here because the inclusion map of a non-open subset of never yields a local homeomorphism.
If is defined by so that geometrically, this map wraps the real line around the circle, then is a local homeomorphism but not a homeomorphism. If is the map that wraps the circle around itself times (i.e. has winding number ), then this is a local homeomorphism for all non-zero but a homeomorphism only in the cases where it is bijective, i.e. when or
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.
There exist local homeomorphisms where is a Hausdorff space and 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.
It is shown in complex analysis that a complex analytic function (where is an open subset of the complex plane ) is a local homeomorphism precisely when the derivative is non-zero for all The function on an open disk around is not a local homeomorphism at when In that case is a point of "ramification" (intuitively, sheets come together there).
Using the inverse function theorem one can show that a continuously differentiable function (where is an open subset of ) is a local homeomorphism if the derivative is an invertible linear map (invertible square matrix) for every (The converse is false, as shown by the local homeomorphism with ). An analogous condition can be formulated for maps between differentiable manifolds.
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 (which is a necessary condition for to be a local homeomorphism) then is a -valued local homeomorphism on a dense open subset of To clarify this statement's conclusion, let be the union of all open subsets of such that the restriction is an injective map, where the assumptions on imply that the (potentially empty[note 1]) set is also the (unique) largest open subset of such that is a local homeomorphism. If every fiber of is a discrete subspace of then is a dense subset of (where being dense in implies, in particular, that if then ). For example, if the continuous open surjection is defined by the polynomial then the maximal open subset from this theorem is which shows that it is possible for to be a proper subset of '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; and if it is not an open map then it is nevertheless still straightforward to apply the theorem (possibly multiple times) by choosing domain(s) based on an appropriate consideration of the map's local minimums/maximums.
A local homeomorphism transfers "local" topological properties in both directions:
- is locally connected if and only if is;
- is locally path-connected if and only if is;
- is locally compact if and only if is;
- is first-countable if and only if is.
As pointed out above, the Hausdorff property is not local in this sense and need not be preserved by local homeomorphisms.
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.
- Consider the continuous open surjection defined by The set for this map is the empty set; that is, there does not exist any non-empty open subset of for which the restriction is an injective map.
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