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. 
If formula_1 is a local homeomorphism, formula_2 is said to be an étale space over formula_3 Local homeomorphisms are used in the study of sheaves. Typical examples of local homeomorphisms are covering maps.
A topological space formula_2 is locally homeomorphic to if every point of formula_2 has a neighborhood that is homeomorphic to an open subset of . 
For example, a manifold of dimension formula_6 is locally homeomorphic to formula_7
If there is a local homeomorphism from formula_2 to formula_9 then formula_2 is locally homeomorphic to formula_9 but the converse is not always true. 
For example, the two dimensional sphere, being a manifold, is locally homeomorphic to the plane formula_12 but there is no local homeomorphism between them (in either direction).
Formal definition.
Let formula_2 and formula_14 be topological spaces. 
A function formula_1 is a local homeomorphism if for every point formula_16 there exists an open set formula_17 containing formula_18 such that the image formula_19 is open in formula_14 and the restriction formula_21 is a homeomorphism (where the respective subspace topologies are used on formula_17 and on formula_19).
Examples and sufficient conditions.
Every homeomorphism is also a local homeomorphism. 
If formula_1 is a local homeomorphism and formula_17 is an open subset of formula_26 then the restriction formula_27 is also a local homeomorphism. The composition of two local homeomorphisms is a local homeomorphism (that is, if formula_1 and formula_29 are local homeomorphisms, then the composition formula_30 is also a local homeomorphism). If formula_1 is continuous, formula_29 is a local homeomorphism, and formula_30 a local homeomorphism, then formula_34 is also a local homeomorphism. 
If formula_17 is an open subset of formula_14 equipped with the subspace topology, then the inclusion map formula_37 is a local homeomorphism. The subset formula_17 being open in formula_2 is essential here because the inclusion map of a non-open subset of formula_14 yields a local homeomorphism. 
If formula_41 is defined by formula_42 so that geometrically, this map wraps the real line around the circle, then formula_41 is a local homeomorphism but not a homeomorphism. 
If formula_44 is the map that wraps the circle around itself formula_6 times (i.e. has winding number formula_6), then this is a local homeomorphism for all non-zero formula_47 but a homeomorphism only in the cases where it is bijective, i.e. when formula_48 or formula_49 
Generalizing the previous two examples, every covering map is a local homeomorphism; in particular, the universal cover formula_50 of a space formula_14 is a local homeomorphism. 
In certain situations the converse is true. For example: if formula_52 is a proper local homeomorphism between two Hausdorff spaces and if formula_14 is also locally compact, then formula_54 is a covering map. 
There exist local homeomorphisms formula_1 where formula_14 is a Hausdorff space and formula_2 is not. 
Consider for instance the quotient space formula_58 where the equivalence relation formula_59 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 formula_60 are not identified and they do not have any disjoint neighborhoods, so formula_2 is not Hausdorff. One readily checks that the natural map formula_62 is a local homeomorphism. 
The fiber formula_63 has two elements if formula_64 and one element if formula_65 
Similarly, it is possible to construct a local homeomorphisms formula_1 where formula_2 is Hausdorff and formula_14 is not: pick the natural map from formula_69 to formula_70 with the same equivalence relation formula_59 as above. 
It is shown in complex analysis that a complex analytic function formula_72 (where formula_17 is an open subset of the complex plane formula_74) is a local homeomorphism precisely when the derivative formula_75 is non-zero for all formula_76 
The function formula_77 on an open disk around formula_60 is not a local homeomorphism at formula_60 when formula_80 
In that case formula_60 is a point of "ramification" (intuitively, formula_6 sheets come together there). 
Using the inverse function theorem one can show that a continuously differentiable function formula_83 (where formula_17 is an open subset of formula_85) is a local homeomorphism if the derivative formula_86 is an invertible linear map (invertible square matrix) for every formula_87 
(The converse is false, as shown by the local homeomorphism formula_88 with formula_89). 
An analogous condition can be formulated for maps between differentiable manifolds. 
Suppose formula_1 is a continuous open surjection between two Hausdorff second-countable spaces where formula_2 is a Baire space and formula_14 is a normal space. Let formula_93 be the union of all open subsets formula_17 of formula_2 such that the restriction formula_27 is an injective map, where the assumptions on formula_34 imply that the (potentially empty) set formula_98 is also the (unique) largest open subset of formula_2 such that formula_100 is a local homeomorphism. If every fiber of formula_34 is a discrete subspace of formula_2 then formula_98 is a dense subset of formula_2 (where formula_98 being dense in formula_2 implies, in particular, that if formula_107 then formula_108). 
For example, if the continuous open surjection formula_109 is defined by the polynomial formula_110 then the maximal open subset formula_93 from this theorem is formula_112 which shows that it is possible for formula_98 to be a subset of formula_34's domain. 
Because every fiber of every non-constant polynomial is finite (and thus a discrete, and even compact, subspace), this example generalizes to such polynomials whenever the mapping induced by it is an open map; and if it is not an open map then it is nevertheless still straightforward to apply the theorem (possibly multiple times) by choosing domain(s) based on an appropriate consideration of the map's local minimums/maximums.
Properties.
Every local homeomorphism is a continuous and open map. A bijective local homeomorphism is therefore a homeomorphism. 
Every fiber of a local homeomorphism formula_1 is a discrete subspace of its domain formula_116 
A local homeomorphism formula_1 transfers "local" topological properties in both directions: 
As pointed out above, the Hausdorff property is not local in this sense and need not be preserved by local homeomorphisms.
The local homeomorphisms with codomain formula_14 stand in a natural one-to-one correspondence with the sheaves of sets on formula_127 this correspondence is in fact an equivalence of categories. Furthermore, every continuous map with codomain formula_14 gives rise to a uniquely defined local homeomorphism with codomain formula_14 in a natural way. All of this is explained in detail in the article on sheaves.
Generalizations and analogous concepts.
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.