Local homeomorphism

In mathematics, more specifically topology, a local homeomorphism is intuitively a function, f, between topological spaces that preserves local structure. Equivalently, one can cover the domain of this function by open sets, such that f restricted to each such open set is a homeomorphism onto its image. In particular, every homeomorphism is a local homeomorphism. The formal definition is given below.

Local homeomorphisms are very important in mathematics, particularly in the theory of manifolds (e.g. differential topology) and algebraic topology. An important example of local homeomorphisms are covering maps. Covering maps are important, because unlike local homeomorphisms, they satisfy the local triviality condition, they induce isomorphisms of particular homotopy groups, and in particular, one can lift any path in the base space of a covering map, to a path in the total space. Although local homeomorphisms are not as strong as covering maps in this respect, they have many important applications in differential topology; namely one uses the notion of a local homeomorphism to define a differentiable manifold.

Formal definition

Let X and Y be topological spaces. A function, $f : X \to Y\,$ is a local homeomorphism, if for every point x in X, there exists an open set U containing x, such that f(U) is open in Y and $f|U : U\to f(U)\,$ is a homeomorphism.

Examples

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 : 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. 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) = 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).

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.

Relation to covering maps

If X and Y are locally compact spaces and p : Y → X is a proper local homeomorphism, then p is a covering map.

See also

• Local diffeomorphism