# Local homeomorphism

In mathematics, more specifically topology, a **local homeomorphism** is intuitively a function, *f*, between topological spaces that preserves local structure.

## Formal definition

Let *X* and *Y* be topological spaces. A function is a **local homeomorphism**^{[1]} 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 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 *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.

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 Haudorff and *Y* is locally compact and Hausdorff and *p* : *X* → *Y* is a proper local homeomorphism, then *p* is a covering map.

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. *n* = 1 or -1.

It is shown in complex analysis that a complex analytic function *f* gives a local homeomorphism precisely when the derivative *f* ′(*z*) is non-zero for all *z* in the domain of *f*. 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).

## Properties

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

A local homeomorphism *f* : *X* → *Y* preserves "local" topological properties:

*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

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.

The local homeomorphisms with codomain *Y* stand in a natural 1-1 correspondence with the sheaves of sets on *Y*. 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.

## See also

## References

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