Local homeomorphism

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In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If f : XY is a local homeomorphism, we also say X is an étale space over Y. Local homeomorphisms provide one approach to the study of sheaves.

Formal definition[edit]

Let X and Y be topological spaces. A function f : XY is a local homeomorphism[1] if for every point x in X there exists an open set U containing x, such that the image f(U) is open in Y and the restriction f|U : Uf(U) is a homeomorphism (where the respective subspace topologies are used on U and on f(U)).

Examples[edit]

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 i : UY is a local homeomorphism. Openness is essential here: the inclusion map of a non-open subset of Y never yields a local homeomorphism.

Let f : RS1 be the map that wraps the real line around the circle (i.e. f(t) = eit for all t ϵ R). This is a local homeomorphism but not a homeomorphism.

Let f : S1S1 be the map that wraps the circle around itself n times (i.e. has winding number n). This is a local homeomorphism for all non-zero n, but a homeomorphism only in the cases where it is bijective, i.e. when n = 1 or -1.

Generalizing the previous two examples, every covering map is a local homeomorphism; in particular, the universal cover p : CY 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 p : XY is a proper local homeomorphism, then p is a covering map.

It is shown in complex analysis that a complex analytic function f : UC (where U is an open subset of the complex plane C) is a local homeomorphism precisely when the derivative f ′(z) is non-zero for all z ϵ U. The function f(z) = zn 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" (intuitively, n sheets come together there).

Using the inverse function theorem one can show that a continuously differentiable function f : URn (where U is an open subset of Rn) is a local homeomorphism if and only if the derivative Dxf is an invertible linear map (invertible square matrix) for every x ϵ U. An analogous criterion can be formulated for maps between differentiable manifolds.

Properties[edit]

Every local homeomorphism is a continuous and open map. A bijective local homeomorphism is therefore a homeomorphism.

A local homeomorphism f : XY preserves "local" topological properties:

It is common however to have a local homeomorphism f : XY where Y is a Hausdorff space and X is not. [Consider for instance the quotient space X = (R ⨿ R)/~ , 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 f : XR is a local homeomorphism. The fiber f −1({y}) has two elements if y≥0 and one element if y<0.]

If f : XY is a local homeomorphism and U is an open subset of X, then the restriction f|U is also a local homeomorphism.

If f : XY and g : YZ are local homeomorphisms, then the composition gf : XZ is also a local homeomorphism.

If f : XY is continuous, g : YZ is a local homeomorphism, and gf : XZ a local homeomorphism, then f is also a local homeomorphism.

The local homeomorphisms with codomain Y stand in a natural 1-1 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[edit]

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.

References[edit]

  1. ^ Munkres, James R. (2000). Topology (2nd ed.). Prentice Hall. ISBN 0-13-181629-2.