Local homeomorphism

Let X and Y be topological spaces. A function f : X → 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 → f(U) is a homeomorphism (where the respective subspace topologies are used on U and on f(U)).

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

There are local homeomorphisms f : X → Y where Y is a Hausdorff space and X is not. Consider for instance the quotient space X = (R ⨿ R)/~ , where the equivalence relation ~ on the disjoint union of two copies of the reals identifies every negative real of the first copy with the corresponding negative real of the second copy. The two copies of 0 are not identified and they do not have any disjoint neighborhoods, so X is not Hausdorff. One readily checks that the natural map f : X → R is a local homeomorphism. The fiber f ⁻¹({y}) has two elements if y ≥ 0 and one element if y < 0.

Similarly, we can construct a local homeomorphisms f : X → Y where X is Hausdorff and Y is not: pick the natural map from X = R ⨿ R to Y = (R ⨿ R)/~ with the same equivalence relation ~ as above.

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 the derivative D_xf is an invertible linear map (invertible square matrix) for every x ϵ U. (The converse is false, as shown by the local homeomorphism f : R → R with f(x)=x³ .) An analogous condition can be formulated for maps between differentiable manifolds.

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

A local homeomorphism f : X → Y transfers "local" topological properties in both directions:

• 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.

As pointed out above, the Hausdorff property is not local in this sense and need not be preserved by local homeomorphisms.

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.

If f : X → Y is continuous, g : Y → Z is a local homeomorphism, and gf : X → Z a local homeomorphism, then f is also a local homeomorphism.

The local homeomorphisms with codomain Y stand in a natural one-to-one correspondence with the sheaves of sets on Y; this correspondence is in fact an equivalence of categories. 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.

The idea of a local homeomorphism can be formulated in geometric settings different from that of topological spaces. For differentiable manifolds, we obtain the local diffeomorphisms; for schemes, we have the formally étale morphisms and the étale morphisms; and for toposes, we get the étale geometric morphisms.