Local homeomorphism

Short description: Mathematical function revertible near each point

In mathematics, more specifically topology, a local homeomorphism is a function between topological spaces that, intuitively, preserves local (though not necessarily global) structure. If [math]\displaystyle{ f : X \to Y }[/math] is a local homeomorphism, [math]\displaystyle{ X }[/math] is said to be an étale space over [math]\displaystyle{ Y. }[/math] Local homeomorphisms are used in the study of sheaves. Typical examples of local homeomorphisms are covering maps.

A topological space [math]\displaystyle{ X }[/math] is locally homeomorphic to [math]\displaystyle{ Y }[/math] if every point of [math]\displaystyle{ X }[/math] has a neighborhood that is homeomorphic to an open subset of [math]\displaystyle{ Y. }[/math] For example, a manifold of dimension [math]\displaystyle{ n }[/math] is locally homeomorphic to [math]\displaystyle{ \R^n. }[/math]

If there is a local homeomorphism from [math]\displaystyle{ X }[/math] to [math]\displaystyle{ Y, }[/math] then [math]\displaystyle{ X }[/math] is locally homeomorphic to [math]\displaystyle{ Y, }[/math] but the converse is not always true. For example, the two dimensional sphere, being a manifold, is locally homeomorphic to the plane [math]\displaystyle{ \R^2, }[/math] but there is no local homeomorphism [math]\displaystyle{ S^2 \to \R^2. }[/math]

Formal definition

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

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 [math]\displaystyle{ \R \to S^1 }[/math] defined by [math]\displaystyle{ t \mapsto e^{it} }[/math] (so that geometrically, this map wraps the real line around the circle) is a local homeomorphism but not a homeomorphism. The map [math]\displaystyle{ f : S^1 \to S^1 }[/math] defined by [math]\displaystyle{ f(z) = z^n, }[/math] which wraps the circle around itself [math]\displaystyle{ n }[/math] times (that is, has winding number [math]\displaystyle{ n }[/math]), is a local homeomorphism for all non-zero [math]\displaystyle{ n, }[/math] but it is a homeomorphism only when it is bijective (that is, only when [math]\displaystyle{ n = 1 }[/math] or [math]\displaystyle{ n = -1 }[/math]).

Generalizing the previous two examples, every covering map is a local homeomorphism; in particular, the universal cover [math]\displaystyle{ p : C \to Y }[/math] of a space [math]\displaystyle{ Y }[/math] is a local homeomorphism. In certain situations the converse is true. For example: if [math]\displaystyle{ p : X \to Y }[/math] is a proper local homeomorphism between two Hausdorff spaces and if [math]\displaystyle{ Y }[/math] is also locally compact, then [math]\displaystyle{ p }[/math] is a covering map.

Local homeomorphisms and composition of functions

The composition of two local homeomorphisms is a local homeomorphism; explicitly, if [math]\displaystyle{ f : X \to Y }[/math] and [math]\displaystyle{ g : Y \to Z }[/math] are local homeomorphisms then the composition [math]\displaystyle{ g \circ f : X \to Z }[/math] 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 [math]\displaystyle{ f : X \to Y }[/math] is a local homeomorphism then its restriction [math]\displaystyle{ f\big\vert_U : U \to Y }[/math] to any [math]\displaystyle{ U }[/math] open subset of [math]\displaystyle{ X }[/math] is also a local homeomorphism.

If [math]\displaystyle{ f : X \to Y }[/math] is continuous while both [math]\displaystyle{ g : Y \to Z }[/math] and [math]\displaystyle{ g \circ f : X \to Z }[/math] are local homeomorphisms, then [math]\displaystyle{ f }[/math] is also a local homeomorphism.

Inclusion maps

If [math]\displaystyle{ U \subseteq X }[/math] is any subspace (where as usual, [math]\displaystyle{ U }[/math] is equipped with the subspace topology induced by [math]\displaystyle{ X }[/math]) then the inclusion map [math]\displaystyle{ i : U \to X }[/math] is always a topological embedding. But it is a local homeomorphism if and only if [math]\displaystyle{ U }[/math] is open in [math]\displaystyle{ X. }[/math] The subset [math]\displaystyle{ U }[/math] being open in [math]\displaystyle{ X }[/math] is essential for the inclusion map to be a local homeomorphism because the inclusion map of a non-open subset of [math]\displaystyle{ X }[/math] never yields a local homeomorphism (since it will not be an open map).

The restriction [math]\displaystyle{ f\big\vert_U : U \to Y }[/math] of a function [math]\displaystyle{ f : X \to Y }[/math] to a subset [math]\displaystyle{ U \subseteq X }[/math] is equal to its composition with the inclusion map [math]\displaystyle{ i : U \to X; }[/math] explicitly, [math]\displaystyle{ f\big\vert_U = f \circ i. }[/math] Since the composition of two local homeomorphisms is a local homeomorphism, if [math]\displaystyle{ f : X \to Y }[/math] and [math]\displaystyle{ i : U \to X }[/math] are local homomorphisms then so is [math]\displaystyle{ f\big\vert_U = f \circ i. }[/math] Thus restrictions of local homeomorphisms to open subsets are local homeomorphisms.

Invariance of domain

Invariance of domain guarantees that if [math]\displaystyle{ f : U \to \R^n }[/math] is a continuous injective map from an open subset [math]\displaystyle{ U }[/math] of [math]\displaystyle{ \R^n, }[/math] then [math]\displaystyle{ f(U) }[/math] is open in [math]\displaystyle{ \R^n }[/math] and [math]\displaystyle{ f : U \to f(U) }[/math] is a homeomorphism. Consequently, a continuous map [math]\displaystyle{ f : U \to \R^n }[/math] from an open subset [math]\displaystyle{ U \subseteq \R^n }[/math] will be a local homeomorphism if and only if it is a locally injective map (meaning that every point in [math]\displaystyle{ U }[/math] has a neighborhood [math]\displaystyle{ N }[/math] such that the restriction of [math]\displaystyle{ f }[/math] to [math]\displaystyle{ N }[/math] is injective).

Local homeomorphisms in analysis

It is shown in complex analysis that a complex analytic function [math]\displaystyle{ f : U \to \Complex }[/math] (where [math]\displaystyle{ U }[/math] is an open subset of the complex plane [math]\displaystyle{ \Complex }[/math]) is a local homeomorphism precisely when the derivative [math]\displaystyle{ f^{\prime}(z) }[/math] is non-zero for all [math]\displaystyle{ z \in U. }[/math] The function [math]\displaystyle{ f(x) = z^n }[/math] on an open disk around [math]\displaystyle{ 0 }[/math] is not a local homeomorphism at [math]\displaystyle{ 0 }[/math] when [math]\displaystyle{ n \geq 2. }[/math] In that case [math]\displaystyle{ 0 }[/math] is a point of "ramification" (intuitively, [math]\displaystyle{ n }[/math] sheets come together there).

Using the inverse function theorem one can show that a continuously differentiable function [math]\displaystyle{ f : U \to \R^n }[/math] (where [math]\displaystyle{ U }[/math] is an open subset of [math]\displaystyle{ \R^n }[/math]) is a local homeomorphism if the derivative [math]\displaystyle{ D_x f }[/math] is an invertible linear map (invertible square matrix) for every [math]\displaystyle{ x \in U. }[/math] (The converse is false, as shown by the local homeomorphism [math]\displaystyle{ f : \R \to \R }[/math] with [math]\displaystyle{ f(x) = x^3 }[/math]). An analogous condition can be formulated for maps between differentiable manifolds.

Local homeomorphisms and fibers

Suppose [math]\displaystyle{ f : X \to Y }[/math] is a continuous open surjection between two Hausdorff second-countable spaces where [math]\displaystyle{ X }[/math] is a Baire space and [math]\displaystyle{ Y }[/math] is a normal space. If every fiber of [math]\displaystyle{ f }[/math] is a discrete subspace of [math]\displaystyle{ X }[/math] (which is a necessary condition for [math]\displaystyle{ f : X \to Y }[/math] to be a local homeomorphism) then [math]\displaystyle{ f }[/math] is a [math]\displaystyle{ Y }[/math]-valued local homeomorphism on a dense open subset of [math]\displaystyle{ X. }[/math] To clarify this statement's conclusion, let [math]\displaystyle{ O = O_f }[/math] be the (unique) largest open subset of [math]\displaystyle{ X }[/math] such that [math]\displaystyle{ f\big\vert_O : O \to Y }[/math] is a local homeomorphism. If every fiber of [math]\displaystyle{ f }[/math] is a discrete subspace of [math]\displaystyle{ X }[/math] then this open set [math]\displaystyle{ O }[/math] is necessarily a dense subset of [math]\displaystyle{ X. }[/math] In particular, if [math]\displaystyle{ X \neq \varnothing }[/math] then [math]\displaystyle{ O \neq \varnothing; }[/math] a conclusion that may be false without the assumption that [math]\displaystyle{ f }[/math]'s fibers are discrete (see this footnote for an example). One corollary is that every continuous open surjection [math]\displaystyle{ f }[/math] between completely metrizable second-countable spaces that has discrete fibers is "almost everywhere" a local homeomorphism (in the topological sense that [math]\displaystyle{ O_f }[/math] is a dense open subset of its domain). For example, the map [math]\displaystyle{ f : \R \to [0, \infty) }[/math] defined by the polynomial [math]\displaystyle{ f(x) = x^2 }[/math] is a continuous open surjection with discrete fibers so this result guarantees that the maximal open subset [math]\displaystyle{ O_f }[/math] is dense in [math]\displaystyle{ \R; }[/math] with additional effort (using the inverse function theorem for instance), it can be shown that [math]\displaystyle{ O_f = \R \setminus \{0\}, }[/math] which confirms that this set is indeed dense in [math]\displaystyle{ \R. }[/math] This example also shows that it is possible for [math]\displaystyle{ O_f }[/math] to be a proper dense subset of [math]\displaystyle{ f }[/math]'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 [math]\displaystyle{ f : X \to Y }[/math] where [math]\displaystyle{ Y }[/math] is a Hausdorff space but [math]\displaystyle{ X }[/math] is not. Consider for instance the quotient space [math]\displaystyle{ X = \left(\R \sqcup \R\right) / \sim, }[/math] where the equivalence relation [math]\displaystyle{ \sim }[/math] 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 [math]\displaystyle{ 0 }[/math] are not identified and they do not have any disjoint neighborhoods, so [math]\displaystyle{ X }[/math] is not Hausdorff. One readily checks that the natural map [math]\displaystyle{ f : X \to \R }[/math] is a local homeomorphism. The fiber [math]\displaystyle{ f^{-1}(\{y\}) }[/math] has two elements if [math]\displaystyle{ y \geq 0 }[/math] and one element if [math]\displaystyle{ y \lt 0. }[/math] Similarly, it is possible to construct a local homeomorphisms [math]\displaystyle{ f : X \to Y }[/math] where [math]\displaystyle{ X }[/math] is Hausdorff and [math]\displaystyle{ Y }[/math] is not: pick the natural map from [math]\displaystyle{ X = \R \sqcup \R }[/math] to [math]\displaystyle{ Y = \left(\R \sqcup \R\right) / \sim }[/math] with the same equivalence relation [math]\displaystyle{ \sim }[/math] 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 [math]\displaystyle{ f : X \to Y }[/math] is a local homeomorphism depends on its codomain. The image [math]\displaystyle{ f(X) }[/math] of a local homeomorphism [math]\displaystyle{ f : X \to Y }[/math] is necessarily an open subset of its codomain [math]\displaystyle{ Y }[/math] and [math]\displaystyle{ f : X \to f(X) }[/math] will also be a local homeomorphism (that is, [math]\displaystyle{ f }[/math] will continue to be a local homeomorphism when it is considered as the surjective map [math]\displaystyle{ f : X \to f(X) }[/math] onto its image, where [math]\displaystyle{ f(X) }[/math] has the subspace topology inherited from [math]\displaystyle{ Y }[/math]). However, in general it is possible for [math]\displaystyle{ f : X \to f(X) }[/math] to be a local homeomorphism but [math]\displaystyle{ f : X \to Y }[/math] to not be a local homeomorphism (as is the case with the map [math]\displaystyle{ f : \R \to \R^2 }[/math] defined by [math]\displaystyle{ f(x) = (x, 0), }[/math] for example). A map [math]\displaystyle{ f : X \to Y }[/math] is a local homomorphism if and only if [math]\displaystyle{ f : X \to f(X) }[/math] is a local homeomorphism and [math]\displaystyle{ f(X) }[/math] is an open subset of [math]\displaystyle{ Y. }[/math]

Every fiber of a local homeomorphism [math]\displaystyle{ f : X \to Y }[/math] is a discrete subspace of its domain [math]\displaystyle{ X. }[/math]

A local homeomorphism [math]\displaystyle{ f : X \to Y }[/math] transfers "local" topological properties in both directions:

• [math]\displaystyle{ X }[/math] is locally connected if and only if [math]\displaystyle{ f(X) }[/math] is;
• [math]\displaystyle{ X }[/math] is locally path-connected if and only if [math]\displaystyle{ f(X) }[/math] is;
• [math]\displaystyle{ X }[/math] is locally compact if and only if [math]\displaystyle{ f(X) }[/math] is;
• [math]\displaystyle{ X }[/math] is first-countable if and only if [math]\displaystyle{ f(X) }[/math] is.

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 [math]\displaystyle{ Y }[/math] stand in a natural one-to-one correspondence with the sheaves of sets on [math]\displaystyle{ Y; }[/math] this correspondence is in fact an equivalence of categories. Furthermore, every continuous map with codomain [math]\displaystyle{ Y }[/math] gives rise to a uniquely defined local homeomorphism with codomain [math]\displaystyle{ Y }[/math] 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.

See also

• Diffeomorphism – Isomorphism of smooth manifolds; a smooth bijection with a smooth inverse
• Homeomorphism – Mapping which preserves all topological properties of a given space
• Isomorphism – In mathematics, invertible homomorphism
• Invariance of domain – Theorem in topology about homeomorphic subsets of Euclidean space
• Local diffeomorphism
• Locally Hausdorff space
• Non-Hausdorff manifold

References

• Bourbaki, Nicolas (1989). General Topology: Chapters 1–4. Éléments de mathématique. Berlin New York: Springer Science & Business Media. ISBN 978-3-540-64241-1. OCLC 18588129. https://doku.pub/documents/31425779-nicolas-bourbaki-general-topology-part-i1pdf-30j71z37920w. 
• Munkres, James R. (2000). Topology (Second ed.). Upper Saddle River, NJ: Prentice Hall, Inc. ISBN 978-0-13-181629-9. OCLC 42683260. 
• Willard, Stephen (2004). General Topology. Dover Books on Mathematics (First ed.). Mineola, N.Y.: Dover Publications. ISBN 978-0-486-43479-7. OCLC 115240.