Local homeomorphism

This article should be merged with étale space.

In topology, a local homeomorphism is a map from one topological space to another that respects locally the topological structure of the two spaces. More precisely, the function f : X → Y is a local homeomorphism if for every point x of X there exists an open neighbourhood U of x and an open set V in Y such that f yields a homeomorphism between U and V.

Some examples

Every homeomorphism is of course also a local homeomorphism, but this is boring.

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.

Taking X and Y to be the circle S¹, regarded as the quotient space R/Z, we can take f to be the function induced by multiplication by n for any integer n. Then 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 and -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).

All covering maps are local homeomorphisms; in particular, the universal cover p : C → X of a space X is a local homeomorphism.

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.