In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If f : X \to Y is a local homeomorphism, X is said to be an étale space over Y. Local homeomorphisms are used in the study of sheaves. Typical examples of local homeomorphisms are covering maps. A topological space X is locally homeomorphic to Y if every point of X has a neighborhood that is homeomorphic to an open subset of Y. For example, a manifold of dimension n is locally homeomorphic to \R^n. If there is a local homeomorphism from X to Y, then X is locally homeomorphic to Y, but the converse is not always true. For example, the two dimensional sphere, being a manifold, is locally homeomorphic to the plane \R^2, but there is no local homeomorphism

Formal definition

A function f : X \to Y between two topological spaces is called a if every point x \in X has an open neighborhood U whose image f(U) is open in Y and the restriction is a homeomorphism (where the respective subspace topologies are used on U and on f(U)).

Examples and sufficient conditions

Local homeomorphisms versus homeomorphisms Every homeomorphism is a local homeomorphism. But a local homeomorphism is a homeomorphism if and only if it is bijective. A local homeomorphism need not be a homeomorphism. For example, the function \R \to S^1 defined by (so that geometrically, this map wraps the real line around the circle) is a local homeomorphism but not a homeomorphism. The map defined by f(z) = z^n, which wraps the circle around itself n times (that is, has winding number n), is a local homeomorphism for all non-zero n, but it is a homeomorphism only when it is bijective (that is, only when n = 1 or n = -1). Generalizing the previous two examples, every covering map is a local homeomorphism; in particular, the universal cover p : C \to Y of a space Y is a local homeomorphism. In certain situations the converse is true. For example: if p : X \to Y is a proper local homeomorphism between two Hausdorff spaces and if Y is also locally compact, then p is a covering map. Local homeomorphisms and composition of functions The composition of two local homeomorphisms is a local homeomorphism; explicitly, if f : X \to Y and g : Y \to Z are local homeomorphisms then the composition is also a local homeomorphism. The restriction of a local homeomorphism to any open subset of the domain will again be a local homomorphism; explicitly, if f : X \to Y is a local homeomorphism then its restriction to any U open subset of X is also a local homeomorphism. If f : X \to Y is continuous while both g : Y \to Z and are local homeomorphisms, then f is also a local homeomorphism. Inclusion maps If is any subspace (where as usual, U is equipped with the subspace topology induced by X) then the inclusion map i : U \to X is always a topological embedding. But it is a local homeomorphism if and only if U is open in X. The subset U being open in X is essential for the inclusion map to be a local homeomorphism because the inclusion map of a non-open subset of X yields a local homeomorphism (since it will not be an open map). The restriction of a function f : X \to Y to a subset is equal to its composition with the inclusion map explicitly, Since the composition of two local homeomorphisms is a local homeomorphism, if f : X \to Y and i : U \to X are local homomorphisms then so is Thus restrictions of local homeomorphisms to open subsets are local homeomorphisms. Invariance of domain Invariance of domain guarantees that if is a continuous injective map from an open subset U of \R^n, then f(U) is open in \R^n and is a homeomorphism. Consequently, a continuous map from an open subset will be a local homeomorphism if and only if it is a locally injective map (meaning that every point in U has a neighborhood N such that the restriction of f to N is injective). Local homeomorphisms in analysis It is shown in complex analysis that a complex analytic function (where U is an open subset of the complex plane \Complex) is a local homeomorphism precisely when the derivative is non-zero for all z \in U. The function f(x) = z^n on an open disk around 0 is not a local homeomorphism at 0 when n \geq 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 (where U is an open subset of \R^n) is a local homeomorphism if the derivative D_x f is an invertible linear map (invertible square matrix) for every x \in U. (The converse is false, as shown by the local homeomorphism with f(x) = x^3). An analogous condition can be formulated for maps between differentiable manifolds. Local homeomorphisms and fibers Suppose f : X \to Y is a continuous open surjection between two Hausdorff second-countable spaces where X is a Baire space and Y is a normal space. If every fiber of f is a discrete subspace of X (which is a necessary condition for f : X \to Y to be a local homeomorphism) then f is a Y-valued local homeomorphism on a dense open subset of X. To clarify this statement's conclusion, let O = O_f be the (unique) largest open subset of X such that is a local homeomorphism. If every fiber of f is a discrete subspace of X then this open set O is necessarily a subset of X. In particular, if then a conclusion that may be false without the assumption that f's fibers are discrete (see this footnote for an example). One corollary is that every continuous open surjection f between completely metrizable second-countable spaces that has discrete fibers is "almost everywhere" a local homeomorphism (in the topological sense that O_f is a dense open subset of its domain). For example, the map defined by the polynomial f(x) = x^2 is a continuous open surjection with discrete fibers so this result guarantees that the maximal open subset O_f is dense in \R; with additional effort (using the inverse function theorem for instance), it can be shown that which confirms that this set is indeed dense in \R. This example also shows that it is possible for O_f to be a dense subset of f'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. Local homeomorphisms and Hausdorffness There exist local homeomorphisms f : X \to Y where Y is a Hausdorff space but X is not. Consider for instance the quotient space where the equivalence relation \sim 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 is a local homeomorphism. The fiber has two elements if y \geq 0 and one element if y < 0. Similarly, it is possible to construct a local homeomorphisms f : X \to Y where X is Hausdorff and Y is not: pick the natural map from to with the same equivalence relation \sim as above.

Properties

A map is a local homeomorphism if and only if it is continuous, open, and locally injective. In particular, every local homeomorphism is a continuous and open map. A bijective local homeomorphism is therefore a homeomorphism. Whether or not a function f : X \to Y is a local homeomorphism depends on its codomain. The image f(X) of a local homeomorphism f : X \to Y is necessarily an open subset of its codomain Y and will also be a local homeomorphism (that is, f will continue to be a local homeomorphism when it is considered as the surjective map onto its image, where f(X) has the subspace topology inherited from Y). However, in general it is possible for to be a local homeomorphism but f : X \to Y to be a local homeomorphism (as is the case with the map defined by for example). A map f : X \to Y is a local homomorphism if and only if is a local homeomorphism and f(X) is an open subset of Y. Every fiber of a local homeomorphism f : X \to Y is a discrete subspace of its domain X. A local homeomorphism f : X \to Y 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 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.

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.

Citations