Let X and Y be topological spaces. A function is a local homeomorphism 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.
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 : 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 Dxf is an invertible linear map (invertible square matrix) for every x ϵ U. An analogous criterion can be formulated for maps between differentiable manifolds.
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.