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Local homeomorphism

In topology, a local homeomorphism is a continuous function f: X \to Y between topological spaces such that for every point p \in X, there exists an open neighborhood U of p on which f restricts to a homeomorphism onto its image f(U), which is open in Y.^[1] Local homeomorphisms preserve local topological structure but may fail to do so globally, distinguishing them from full homeomorphisms. They are necessarily open maps, meaning the image of any open set in X is open in Y, and they are always continuous by definition.^[1] A key property is their role in covering spaces: every covering map is a local homeomorphism, and if a local homeomorphism is also bijective, it is a homeomorphism.^[1] Prominent examples include the projection p: \mathbb{R} \to S^1 defined by p(t) = (\cos(2\pi t), \sin(2\pi t)), which wraps the real line onto the circle, and the n-fold wrapping maps p_n: S^1 \to S^1 given by p_n(\cos(2\pi t), \sin(2\pi t)) = (\cos(2\pi n t), \sin(2\pi n t)) for integer n \neq 0.^[2] These illustrate how local homeomorphisms often arise in constructions of covering spaces and in manifold theory, where charts provide local homeomorphisms to Euclidean space.^[1]

Definition

Formal Definition

A homeomorphism between topological spaces X and Y is a bijective continuous map f: X \to Y whose inverse f^{-1}: Y \to X is also continuous.^[3] In topological spaces, open sets form a basis for the topology, consisting of subsets whose preimages under continuous maps preserve the open structure.^[3] A continuous function f: X \to Y between topological spaces is a local homeomorphism if, for every point x \in X, there exists an open neighborhood U of x such that the restriction f|_U: U \to f(U) is a homeomorphism onto its image f(U), and f(U) is open in Y.^[4] This condition ensures that f preserves local topological structure at every point, with the openness of f(U) guaranteeing that the image inherits the necessary topological properties for the restriction to qualify as a homeomorphism.^[5] The notation f|_U denotes the restriction of f to the subspace U, emphasizing the local bijection and continuity of both f|_U and its inverse (f|_U)^{-1}: f(U) \to U.^[4]

Equivalent Characterizations

A continuous map f: X \to Y between topological spaces is a local homeomorphism if and only if it is an open map and locally injective, meaning that for every x \in X, there exists an open neighborhood U of x such that the restriction f|_U: U \to f(U) is injective. Since f is open, f(U) is open in Y, and the bijectivity onto the image combined with continuity and openness ensures that f|_U is a homeomorphism onto f(U).^[1] Local homeomorphisms are always open maps. To see this, let V \subseteq X be open. For each x \in V, there exists an open neighborhood U_x \subseteq V of x such that f(U_x) is open in Y. Then f(V) = \bigcup_{x \in V} f(U_x) is a union of open sets, hence open in Y. This holds without additional assumptions like Hausdorff spaces on X or Y.^[6]^[1] While global homeomorphisms require bijectivity (with continuous inverse) across the entire spaces, local homeomorphisms emphasize this structure only in neighborhoods of points, allowing non-bijective maps like covering projections (e.g., the infinite-sheeted cover \mathbb{R} \to S^1) that preserve local topology without global invertibility. This locality distinguishes them, as bijectivity alone does not guarantee a local homeomorphism unless accompanied by openness and continuity.^[1]

Examples and Conditions

Illustrative Examples

The identity map on any topological space X to itself serves as the simplest example of a local homeomorphism, since it is a homeomorphism and thus maps every neighborhood of a point homeomorphically onto itself, satisfying the local condition trivially.^[7] A fundamental non-trivial example is the exponential map \exp: \mathbb{C} \to \mathbb{C} \setminus \{0\}, defined by \exp(z) = e^z, which acts as the universal covering map of the punctured complex plane. For any point z_0 \in \mathbb{C}, a small disk neighborhood around z_0 is mapped homeomorphically by \exp onto a slit neighborhood in \mathbb{C} \setminus \{0\}, avoiding the branch cut along the positive real axis to ensure the local inverse exists and is continuous.^[7] Projection maps in covering spaces provide further illustrations, such as the infinite-sheeted covering \pi: \mathbb{R} \to S^1 given by \pi(t) = e^{2\pi i t}, where S^1 is the unit circle in the complex plane. Around any t_0 \in \mathbb{R}, an open interval neighborhood maps homeomorphically onto an open arc on S^1, with the preimage of that arc consisting of disjoint translates of the interval, each mapped bijectively.^[1] Constant maps, which send every point in the domain to a fixed point in the codomain, fail as local homeomorphisms because any neighborhood in the domain maps to a singleton, which is not homeomorphic to an open set in the codomain unless the spaces are indiscrete. Likewise, continuous bijections that are not open—such as the inclusion of a non-open subset where local neighborhoods do not map onto open sets—do not qualify, as they violate the requirement for local surjectivity onto open neighborhoods.^[7] Early examples of local homeomorphisms emerged in the 19th century through Bernhard Riemann's 1851 doctoral dissertation, where he introduced Riemann surfaces to resolve multi-valued analytic functions, constructing surfaces that are locally homeomorphic to open sets in the complex plane via charts that preserve the structure of the domain.^[8]

Sufficient Conditions for Local Homeomorphisms

One standard sufficient condition for a continuous map f: X \to Y between topological spaces to be a local homeomorphism is that it is open and locally injective, provided the spaces satisfy mild regularity assumptions such as being locally compact Hausdorff. In this setting, the openness ensures that images of suitable neighborhoods are open in Y, while local injectivity guarantees bijectivity onto those images, and the continuity of f combined with the topological properties of the spaces ensures the local inverses are continuous, making f a homeomorphism onto each such image. This criterion is particularly useful in manifold theory and algebraic geometry, where such spaces are common.^[9] In the context of differentiable maps between manifolds or Euclidean spaces, which are metric spaces, a map f: U \subset \mathbb{R}^n \to \mathbb{R}^n is a local homeomorphism if it is C^1 and the Jacobian determinant does not vanish at any point in U, meaning the derivative Df(x) is invertible for all x \in U. By the inverse function theorem, around each point x, there exist neighborhoods where f is a diffeomorphism onto its image, hence a homeomorphism, establishing the local homeomorphism property. This condition extends to everywhere differentiable maps (not necessarily C^1) with invertible derivatives everywhere.^[10] For maps on the real line, a continuous strictly monotonic function f: I \to \mathbb{R}, where I is an open interval, is a local homeomorphism, as it is a homeomorphism onto its image, which is open in \mathbb{R}. Continuous strictly monotone functions between intervals are homeomorphisms onto their images.^[11] An important related criterion arises when considering proper maps: if f: X \to Y is a local homeomorphism that is also proper (meaning preimages of compact sets are compact) between nice spaces, such as locally path-connected locally compact Hausdorff spaces, then f satisfies the evenly covered condition, making it a covering map; however, properness alone does not suffice for local homeomorphism without additional assumptions like openness or local injectivity.^[1]

Properties

Composition and Stability

Local homeomorphisms exhibit stability under composition. Specifically, if f: X \to Y and g: Y \to Z are local homeomorphisms between topological spaces, then the composition g \circ f: X \to Z is also a local homeomorphism.^[6] To verify this, consider a point x \in X. There exists an open neighborhood U \subset X containing x such that f(U) is open in Y and f|_U: U \to f(U) is a homeomorphism. Let y = f(x); then there exists an open neighborhood V \subset Y containing y such that g(V) is open in Z and g|_V: V \to g(V) is a homeomorphism. The set W = U \cap f^{-1}(V) is an open neighborhood of x in X, f(W) is open in V (hence in Y), and f|_W: W \to f(W) is a homeomorphism. Moreover, (g \circ f)|_W = g|_{f(W)} \circ f|_W, which is a composition of homeomorphisms, so it is a homeomorphism onto the open set g(f(W)) in Z.^[6] Local homeomorphisms are also stable under restriction to open subspaces. If f: X \to Y is a local homeomorphism and A \subset X is open, then the restriction f|_A: A \to Y is a local homeomorphism, where A inherits the subspace topology from X.^[6] For a point a \in A, since A is open in X, any open neighborhood U \subset X of a intersects A in an open set U \cap A in the subspace topology on A. The local homeomorphism property of f ensures that f(U \cap A) is open in Y and f|_{U \cap A} is a homeomorphism onto its image.^[12] By definition, every local homeomorphism admits local inverses: for each x \in X, there exists an open neighborhood U \subset X such that f(U) is open in Y and f|_U has a continuous inverse defined on f(U).^[12] These local inverses exist pointwise but do not necessarily extend to a global continuous inverse on Y, distinguishing local homeomorphisms from global homeomorphisms. If f is in fact a global homeomorphism, then its global inverse coincides with the local inverses on the corresponding neighborhoods.^[6] However, the class of local homeomorphisms is not closed under composition with arbitrary continuous maps. For a counterexample, consider the continuous map k: \mathbb{R} \to \mathbb{R} given by k(x) = 0 if x \leq 0 and k(x) = x if x > 0. This is not a local homeomorphism at x = 0, since any neighborhood (- \delta, \delta) maps onto [0, \delta) but k|_{(- \delta, \delta)} fails to be bijective onto its image (as all points in (-\delta, 0] map to 0). Composing k with the identity map \mathrm{id}: \mathbb{R} \to \mathbb{R}, which is a local homeomorphism, yields k \circ \mathrm{id} = k, which is not a local homeomorphism.

Preserved Topological Properties

Local homeomorphisms preserve local connectedness because, for each point in the domain, there exists a neighborhood that is homeomorphic to its image under the map, and homeomorphisms preserve connectedness of open sets.^[13] Similarly, local path-connectedness is preserved, as paths within such neighborhoods are mapped homeomorphically, maintaining the existence of paths between nearby points.^[13] Local compactness is also preserved, since the compact neighborhoods in the domain are mapped to compact open sets in the codomain via the local homeomorphisms, which are open maps.^[14] However, global topological properties such as connectedness or compactness are not necessarily preserved. For instance, the exponential map from \mathbb{R} to S^1, defined by t \mapsto e^{2\pi i t}, is a local homeomorphism that preserves local connectedness and compactness but maps the non-compact, simply connected \mathbb{R} onto the compact, non-simply connected circle S^1. Local homeomorphisms induce isomorphisms on local homology groups. Specifically, if f: X \to Y is a local homeomorphism at x \in X, then f induces an isomorphism f_*: H_n(X, X \setminus \{x\}) \to H_n(Y, Y \setminus \{f(x)\}) for all n, preserving these local topological invariants.^[15] In the context of manifolds, local homeomorphisms preserve the local dimension, as the isomorphism on local homology groups ensures that the dimension n at a point—characterized by H_n(M, M \setminus \{p\}) \cong \mathbb{Z} and vanishing in other degrees—remains unchanged.^[15]

Relation to Sheaves and Local Triviality

A local homeomorphism f: X \to Y between topological spaces induces a natural sheaf of sections over Y, defined on an open set U \subseteq Y as the set \Gamma(U, f) of continuous sections s: U \to X such that f \circ s = \mathrm{id}_U, equipped with the obvious restriction maps.^[16] This construction yields an étale sheaf, as the associated étale space is precisely f itself, which is a local homeomorphism, and the sheaf satisfies the gluing axiom due to the local invertibility of f.^[17] In this context, the sheaf \Gamma(-, f) captures the local bijectivity of f, transforming the topological mapping into an algebraic structure that encodes how local inverses glue together over open covers of Y. A fundamental theorem states that this sheaf of local inverses (or sections) forms a sheaf of sets on Y, where the stalks at each point y \in Y are discrete spaces isomorphic to the fiber f^{-1}(y) equipped with the discrete topology induced by the local homeomorphism property.^[18] Specifically, the stalk \Gamma_y(f) consists of germs of sections near y, each corresponding to a unique point in the fiber, and the local homeomorphism ensures that these germs do not accumulate, rendering the stalk discrete.^[19] This discreteness reflects the étale nature of the sheaf, distinguishing it from more general sheaves where stalks may carry additional topological structure. The map f exhibits local triviality when its fibers are discrete and have uniform cardinality over small neighborhoods in Y, meaning that locally, f resembles a product bundle with discrete fiber, akin to the structure of covering spaces.^[17] In such cases, the induced sheaf \Gamma(-, f) is locally constant, with sections over contractible opens being constant functions to the fiber set. This property bridges to covering space theory, where finite-sheeted local homeomorphisms with discrete uniform fibers yield regular coverings.^[18] This interplay between local homeomorphisms and sheaves originated in the mid-1950s through Alexander Grothendieck's foundational work in algebraic geometry, where he modeled étale morphisms topologically as local homeomorphisms to develop the étale topology and cohomology.^[17] In publications like Revêtements étales et groupe de fondamental (SGA 1, 1960–1961), Grothendieck used these ideas to generalize classical topology to schemes, treating étale maps as "local isomorphisms" whose sheaf-theoretic duals facilitate computations in arithmetic geometry.

Applications and Relations

Role in Covering Spaces and Manifolds

Local homeomorphisms are central to the theory of covering spaces in algebraic topology. A key result states that if p: E \to B is a local homeomorphism and the base space B is locally path-connected and semilocally simply connected, then p is a covering map.^[3] This theorem ensures that under these conditions on B, the local triviality provided by the homeomorphisms over small neighborhoods extends globally to the evenly covered property defining covering maps.^[3] Such spaces B admit universal covers, and the result facilitates the classification of all connected coverings up to isomorphism via subgroups of the fundamental group.^[3] In differential geometry, local homeomorphisms underpin the construction of manifolds. A topological manifold is defined using an atlas of charts, where each chart consists of an open set U \subset [M](/page/M) and a homeomorphism \phi: U \to V \subset \mathbb{R}^n to an open subset of the model space \mathbb{R}^n.^[20] Compatibility between charts requires that transition maps \phi_j \circ \phi_i^{-1}, defined on overlaps, are homeomorphisms between open sets in \mathbb{R}^n.^[20] This structure implies that [M](/page/M) is covered by open sets each homeomorphic to portions of the model space, with the atlas ensuring a consistent topological gluing via these local homeomorphisms.^[20] Consequently, the manifold itself serves as a total space that locally covers the model space through its charts, though globally it may exhibit non-trivial topology. In orbifold theory, the role of local homeomorphisms in handling singularities is highlighted. An orbifold is a space locally modeled by quotients \mathbb{R}^n / G, where G is a finite group of diffeomorphisms, and orbifold charts consist of homeomorphisms from open sets in these quotient spaces to open sets in the orbifold.^[21] This framework extends manifold definitions by allowing effective singularities, with the homeomorphisms ensuring that the space behaves like a manifold away from singular strata while incorporating group actions in local models.^[21] Such constructions enable the study of orbifolds as branched covers of manifolds, bridging singular geometry with classical covering theory.^[21]

Connections to Other Mappings

Local homeomorphisms are closely related to local diffeomorphisms in the context of smooth manifolds. A local diffeomorphism is the smooth analogue of a local homeomorphism, defined as a smooth map between manifolds whose differential is an isomorphism at every point, thereby preserving the tangent space structure locally.^[22] This ensures that the map is a local homeomorphism on the underlying topological spaces while additionally respecting the differentiable structure.^[22] In the differentiable category, local homeomorphisms correspond to submersions with discrete fibers. Specifically, a smooth map that is a local homeomorphism must have an invertible differential everywhere, making it a submersion, and its fibers must be discrete to maintain the local bijectivity required for homeomorphism onto images.^[23] This characterization highlights how the topological notion of local homeomorphism imposes constraints on the differential behavior when smoothness is added. Étale maps provide an algebraic geometry perspective analogous to local homeomorphisms in topology. An étale morphism between schemes is flat and unramified, serving as the direct counterpart to a local homeomorphism by locally resembling isomorphisms in the étale topology, much like how local homeomorphisms preserve local topological structure.^[17] This analogy extends the topological concept to algebraic settings, where étale maps facilitate the study of coverings and cohomology in a manner parallel to topological coverings. A refinement in modern homotopy theory involves quasi-étale maps, which are local homeomorphisms equipped with finite fibers, bridging étale morphisms and finite covers. These maps, defined in analytic and scheme contexts as quasi-finite and étale in codimension one, adapt the local homeomorphism property to scenarios with bounded fiber cardinality, enhancing applications in étale homotopy types without altering key topological invariants.^[24] ^[25]

Generalizations

In Other Topological Categories

In the category of uniform spaces, local uniform homeomorphisms extend the notion of local homeomorphisms by requiring preservation of the uniform structure locally. Specifically, a map f: (X, \mathcal{U}_X) \to (Y, \mathcal{U}_Y) between uniform spaces is a local uniform homeomorphism if it is continuous and, for every point x \in X, there exists a uniform neighborhood U of x (defined by an entourage in \mathcal{U}_X) such that f|_U: U \to f(U) is a uniform homeomorphism onto its image, meaning both f|_U and its inverse are uniformly continuous.^[26] This ensures that the map respects not only the underlying topology but also the uniformity, which controls notions like Cauchy sequences and completeness locally.^[27] In the setting of locales, which formalize pointless topology via frames of open sets, local homeomorphisms are defined in terms of frame homomorphisms. A locale map f: X \to Y, represented by a frame homomorphism f^*: \Omega Y \to \Omega X, is a local homeomorphism if f is an open map (i.e., the direct image f_! preserves finite meets) and the relative diagonal \Delta_f: X \to X \times_Y X is an open embedding.^[28] Equivalently, using generalized points, every generalized point of X has an open neighborhood where f restricts to an isomorphism of locales.^[29] This construction aligns local homeomorphisms with étale maps in pointless settings, where sheaves over a locale correspond to local homeomorphisms with discrete fibers, differing from spatial topology by avoiding reliance on points and emphasizing lattice-theoretic structure.^[30] A key result in categorical topology states that, in the category of compactly generated spaces (also known as k-spaces), the local homeomorphisms coincide with those defined in the full category of topological spaces. This holds because the compactly generated topology is the finest topology making all maps from compact Hausdorff spaces continuous, ensuring that local properties, such as the existence of homeomorphic neighborhoods, are detected precisely on compact subsets without altering the underlying local structure.^[31] In homotopy categories, the analogue of local homeomorphisms appears as local weak equivalences, which generalize the concept to homotopical settings by requiring maps to induce weak equivalences (isomorphisms on homotopy groups) at every stalk or local level. For instance, in the model category of simplicial presheaves on a site, a local weak equivalence X \to Y is a map such that, for every point in the base site, the induced map on stalks is a weak homotopy equivalence.^[32] These equivalences form the weak equivalences in localized model structures for local homotopy theory, inducing isomorphisms on homology sheaves and cohomology groups, and provide a framework for studying local homotopical properties that standard homotopy equivalences overlook, such as in étale or motivic homotopy.^[33]

Analogous Concepts in Geometry and Algebra

In differential geometry, local diffeomorphisms provide the smooth analogue of local homeomorphisms between topological spaces. A smooth map f: M \to N between smooth manifolds is a local diffeomorphism if, for every point p \in M, there exist neighborhoods U \subset M of p and V \subset N of f(p) such that f restricts to a diffeomorphism from U onto V. This condition is equivalent to the differential df_p: T_p M \to T_{f(p)} N being an isomorphism for every p, as guaranteed by the inverse function theorem, ensuring that f is a local homeomorphism with smooth inverses.^[22] Local diffeomorphisms preserve the differential structure locally and play a key role in coordinate charts and atlases for manifolds, where they facilitate the transition between local Euclidean coordinates. In algebraic geometry, étale morphisms generalize the notion of local homeomorphisms to the setting of schemes, adapting it to the Zariski topology and beyond. A morphism of schemes f: X \to Y is étale if it is flat, unramified, and locally of finite presentation; this makes f locally isomorphic in the étale topology, which refines the Zariski topology to mimic classical covering spaces.^[34] Étale morphisms thus serve as the algebraic counterpart to local diffeomorphisms or homeomorphisms, enabling the study of Galois theory and cohomology via étale covers, where the flatness ensures proper fiber behavior and finite presentation controls complexity.^[17] Seminal work by Grothendieck established this framework, highlighting étale maps as "local isomorphisms" in algebraic contexts.^[17] In complex analysis, particularly on Riemann surfaces, holomorphic local homeomorphisms correspond to unbranched holomorphic coverings, where the map is locally biholomorphic without critical points. A holomorphic map f: X \to Y between Riemann surfaces is a local homeomorphism if and only if f'(z) \neq 0 everywhere, making it an open mapping that locally inverts via the inverse function theorem for holomorphic functions. Such maps form unbranched coverings, preserving the complex structure and allowing analytic continuation along paths, as in the universal cover of the punctured plane by the exponential map.^[35] Emerging analogies in derived geometry, particularly in the 2020s, extend local homeomorphisms to model the stacky structures of derived stacks and moduli spaces, where they describe local presentations amid homotopical singularities. In this framework, derived morphisms that behave as local homeomorphisms on underlying topological spaces capture the "étale-like" local equivalences in higher categorical geometry, facilitating representability criteria for elliptic moduli problems and pseudo-holomorphic curve spaces. These developments, building on Lurie's derived algebraic geometry, use local homeomorphisms to resolve ambiguities in singular loci, providing interdisciplinary bridges to topological and smooth settings.

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Below is a merged response summarizing all mentions of "local homeomorphism" in Hatcher’s *Algebraic Topology* (https://pi.math.cornell.edu/~hatcher/AT/AT.pdf). The information is consolidated into a detailed narrative with a table for structured data where applicable, ensuring all details from the individual segments are retained. Since the system has a "no thinking token" limit, I’ll focus on directly compiling and organizing the provided information efficiently.
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Maps that admit local sections through each 'point' in the domain
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Definition 3.3 (Quasi-étale morphisms). A morphism f : X → Y between normal varieties is called quasi-étale if f is quasi-finite and étale in codimension one.<|control11|><|separator|>
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### Summary of Local Homeomorphisms in Point-Free Topology (Locales)
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