# Local homeomorphism

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* : *X* → *Y* is a local homeomorphism, *X* is said to be an **étale space** over *Y.* Local homeomorphisms are used in the study of sheaves. Typical examples of local homeomorphisms are covering maps.

A topological space X is **locally homeomorphic** to Y if every point of X has a neighborhood that is homeomorphic to an open subset of Y. For example, a manifold of dimension n is locally homeomorphic to

If there is a local homeomorphism from X to Y, then X is locally homeomorphic to Y, 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).

## Formal definition

Let *X* and *Y* be topological spaces. A function *f* : *X* → *Y* 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}* :

*U*→

*f*(

*U*) is a homeomorphism (where the respective subspace topologies are used on

*U*and on

*f*(

*U*)).

## 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 *i* : *U* → *Y* 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* : **R** → *S*^{1} be the map that wraps the real line around the circle (i.e. *f*(*t*) *= e ^{it}* for all

*t*ϵ

**R**). This is a local homeomorphism but not a homeomorphism.

Let *f* : *S*^{1} → *S*^{1} 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* : *C* → *Y* 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* : *X* → *Y* is a proper local homeomorphism, then *p* is a covering map.

There are local homeomorphisms *f* : *X* → *Y* 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* : *X* → **R** is a local homeomorphism. The fiber *f* ^{−1}({*y*}) has two elements if *y* ≥ 0 and one element if *y* < 0.

Similarly, we can construct a local homeomorphisms *f* : *X* → *Y* where *X* is Hausdorff and *Y* is not: pick the natural map from *X* = **R** ⨿ **R** to Y = (**R** ⨿ **R**)/~ with the same equivalence relation ~ as above.

It is shown in complex analysis that a complex analytic function *f* : *U* → **C** (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*) = *z*^{n} 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* : *U* → **R**^{n} (where *U* is an open subset of **R**^{n}) is a local homeomorphism if the derivative D_{x}*f* is an invertible linear map (invertible square matrix) for every x ϵ *U.* (The converse is false, as shown by the local homeomorphism *f* : **R** → **R** with *f*(*x*)=*x*^{3} .) 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 *f* : *X* → *Y* transfers "local" topological properties in both directions:

*X*is locally connected if and only if*f*(*X*) is;*X*is locally path-connected if and only if*f*(*X*) is;*X*is locally compact if and only if*f*(*X*) is;*X*is first-countable if and only if*f*(*X*) is.

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

If *f* : *X* → *Y* is a local homeomorphism and *U* is an open subset of *X*, then the restriction *f*|_{U} is also a local homeomorphism.

If *f* : *X* → *Y* and *g* : *Y* → *Z* are local homeomorphisms, then the composition *gf* : *X* → *Z* is also a local homeomorphism.

If *f* : *X* → *Y* is continuous, *g* : *Y* → *Z* is a local homeomorphism, and *gf* : *X* → *Z* 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.

## References

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