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.

Formal definition

Let X and Y be topological spaces. A function $f:X\to Y\,$ 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 $f|_{U}:U\to f(U)\,$ 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.

Let f : R → S¹ be the map that wraps the real line around circle (i.e. f(t) = e^it for all t ϵ R). This is a local homeomorphism but not a homeomorphism.

Let f : S¹ → S¹ 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 Haudorff and Y is locally compact and Hausdorff and p : X → Y is a proper local homeomorphism, then p is a covering map.

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) = 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 : U → Rⁿ (where U is an open subset of Rⁿ) is a local homeomorphism if and only if the derivative D_xf 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

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

• Local diffeomorphism

References

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