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Compact-open topology

The compact-open topology is a topology defined on the set C(X, Y) of all continuous maps from a topological space X to a topological space Y, generated as the coarsest topology making the evaluation map ev: X \times C(X, Y) \to Y, given by ev(x, f) = f(x), continuous whenever X is locally compact and Hausdorff.^[1] Its subbasis consists of all sets of the form V(K, U) = \{f \in C(X, Y) \mid f(K) \subseteq U\}, where K \subseteq X is compact and U \subseteq Y is open.^[2] This topology was introduced by Ralph H. Fox in 1945 to address the problem of topologizing function spaces in a way that equates the continuity of a joint map h: X \times T \to Y with the continuity of its curried associate h^*: T \to C(X, Y).^[1] Fox proved that, under conditions such as X being regular and locally compact, the compact-open topology achieves this equivalence for arbitrary T.^[1] The construction was further developed by Richard Arens in 1946, who extended its properties and applications.^[3] In algebraic topology, the compact-open topology is essential for studying mapping spaces C(X, Y), as it endows them with a structure that supports homotopy theory; for instance, it makes composition maps continuous when X and Y are compactly generated Hausdorff spaces, facilitating the definition of homotopy groups via path and loop spaces.^[4] When Y is a metric space, this topology coincides with the topology of uniform convergence on compact subsets of X, ensuring that sequences of functions converge pointwise on compacts if and only if they converge uniformly there.^[2] It also appears prominently in fiber bundle theory, where it topologies the space of sections or transition functions, as in Norman Steenrod's foundational treatment of bundles. Key properties include that the compact-open topology on C(X, Y) is Hausdorff if Y separates points, and it preserves exponentials in the category of topological spaces under restrictions like local compactness, making the category cartesian closed.^[5] These features underpin its use in advanced topics, such as the homotopy type of function spaces between CW-complexes and the construction of universal covering spaces via path space fibrations.^[4]

Definition and Motivation

Formal Definition

Let X and Y be topological spaces. The set C(X, Y) consists of all continuous functions (or maps) from X to Y.^[1] The compact-open topology on C(X, Y) is generated by a subbasis consisting of sets of the form [K, V] = \{f \in C(X, Y) \mid f(K) \subseteq V\}, where K ranges over all compact subsets of X and V ranges over all open subsets of Y.^[1] The role of compactness for subsets of X is essential in this construction, as it ensures the subbasis elements capture uniform behavior of functions over "small" portions of the domain in a topological sense.^[1] A basis for the compact-open topology is given by the finite intersections of these subbasis elements; specifically, for finitely many pairs (K_i, V_i) with i = 1, \dots, n, the set \bigcap_{i=1}^n [K_i, V_i] = \{f \in C(X, Y) \mid f(K_i) \subseteq V_i \ \forall i\}.^[1] This defines the coarsest topology on C(X, Y) (i.e., the smallest topology containing the subbasis), making the compact-open topology the initial topology induced by the evaluation maps at compact sets.^[1]

Historical Context and Motivation

The compact-open topology was first introduced by Ralph H. Fox in 1945, within the framework of algebraic topology, as a means to endow the space of continuous functions between topological spaces with a suitable topology.^[1] Motivated by a challenge posed by Witold Hurewicz, Fox sought to resolve the limitations of prior approaches that required the domain space to be locally compact—a condition that excluded many relevant function spaces, such as those arising in homotopy theory.^[1] His construction ensured that the induced topology on the function space preserved key continuity properties, such as the equivalence between joint continuity of a map and the continuity of its associated currying, under weaker assumptions like first countability.^[1] Independently, Richard F. Arens developed the topology in 1946, focusing on its application to spaces of transformations and homeomorphisms, where it provided a natural structure for studying groups of continuous maps.^[6] Arens emphasized the topology's role in making the set of continuous functions into a topological space that supports algebraic operations, such as composition, while highlighting its utility for general function spaces beyond specific algebraic topology contexts.^[6] The primary motivation for the compact-open topology stemmed from the need to define a topology on the space C(X, Y) of continuous maps from a topological space X to Y such that the evaluation map C(X, Y) \times X \to Y, given by (f, x) \mapsto f(x), is continuous.^[1] This construction also ensured that compactness in mapping spaces is preserved when X is locally compact, allowing the exponential law—identifying maps from Z \times X to Y with maps from Z to C(X, Y)—to hold topologically.^[1] Furthermore, it generalized the notion of uniform convergence to settings where Y lacks a metric, by inducing convergence uniform on compact subsets of X.^[1] The compact-open topology plays a crucial role in the study of Pontryagin duality for locally compact abelian groups, where it equips the dual group of continuous homomorphisms with a compatible topology that facilitates the duality theorem.^[7]

Properties

Topological Properties

The compact-open topology on the space C(X, Y) of continuous functions from a topological space X to a topological space Y is Hausdorff whenever Y is Hausdorff.^[8] This follows from the subbasis elements V(\{x\}, U) = \{f \in C(X, Y) \mid f(x) \in U\} for x \in X and open U \subseteq Y, which separate distinct functions f and g at points where f(x) \neq g(x), as singletons are compact subsets of X.^[8] The compact-open topology makes all the point evaluation maps \mathrm{ev}_x: C(X, Y) \to Y continuous and refines (is finer than) the topology of pointwise convergence, which is the initial topology induced by these maps.^[9] For X locally compact and Y a Hausdorff uniform space, a variant of the Arzelà-Ascoli theorem states that if K \subseteq C(X, Y) is equicontinuous and pointwise relatively compact (meaning \{f(x) \mid f \in K\} is relatively compact in Y for each x \in X), and satisfies an extension property, then K is relatively compact in the compact-open topology.^[10] Equicontinuity ensures uniform control on compact subsets of X, aligning with the subbasis of the topology. The compact-open topology is metrizable when X is locally compact Hausdorff and Y is a metric space. In this setting, a compatible metric can be constructed using a countable exhaustion of X by compact sets K_n, defined as

d(f, g) = \sum_{n=1}^\infty 2^{-n} \min\left(1, \sup_{x \in K_n} d_Y(f(x), g(x))\right),

which generates the topology via uniform convergence on the K_n.^[11] Compactness is preserved in the compact-open topology under suitable conditions on X and subsets of Y; for instance, if X is compact Hausdorff and K \subseteq Y is compact, then the subspace \{f \in C(X, Y) \mid f(X) \subseteq K\} is compact, as the compact-open topology coincides with the uniform topology on this bounded set.^[11]

Convergence and Continuity

In the compact-open topology on the space C(X,Y) of continuous functions from a topological space X to a topological space Y, a net (f_\alpha) in C(X,Y) converges to f \in C(X,Y) if and only if, for every compact subset K \subseteq X, the net (f_\alpha) converges to f uniformly on K. This characterization highlights the topology's emphasis on controlled behavior on compact sets, distinguishing it from weaker forms of convergence. For sequences, the same criterion applies when X is first countable, but nets provide the general framework for convergence in non-metrizable settings. When Y admits a compatible uniform structure (i.e., Y is uniformizable), the compact-open topology coincides precisely with the topology induced by the uniform structure of uniform convergence on compact subsets of X. This equivalence ensures that the topology is uniformizable itself under these conditions, allowing for the study of Cauchy nets and completeness in a uniform sense. In contrast, if Y lacks a uniform structure, the compact-open topology remains defined via its subbasis but may not align directly with a uniform convergence notion. The compact-open topology supports continuity of key operations on function spaces. Specifically, if Y is locally compact, the composition map C(Y,Z) \times C(X,Y) \to C(X,Z), given by (g,f) \mapsto g \circ f, is continuous with respect to the product topology on the domain and the compact-open topology on the codomain, for topological spaces X,Y,Z. This joint continuity facilitates algebraic structures like monoids on mapping spaces. Additionally, the evaluation map \mathrm{ev}: X \times C(X,Y) \to Y, defined by (x,f) \mapsto f(x), is continuous when X is locally compact, X carries its given topology, C(X,Y) the compact-open topology, and Y its topology; this follows from the subbasis elements when compact neighborhoods cover points in X. The compact-open topology differs from the topology of pointwise convergence, which is strictly coarser unless X is discrete, as pointwise convergence requires only convergence at each point without uniformity on compacts. It is also coarser than the topology of uniform convergence on all of X, unless X itself is compact, in which case the two coincide. These distinctions underscore the compact-open topology's intermediate role, balancing local control with global function space structure.

Applications

In Homotopy Theory

In homotopy theory, the compact-open topology equips the mapping space \Map(X, Y), consisting of continuous maps from a topological space X to Y, with a structure that models the homotopy type of the set of homotopy classes [X, Y]. This topology ensures that \Map(X, Y) captures essential homotopical information, such as path components corresponding to homotopy classes of maps, making it a fundamental tool for studying spaces up to homotopy equivalence. For instance, when X and Y are CW-complexes, the compact-open topology on \Map(X, Y) makes it homotopy equivalent to a CW-complex under mild conditions, facilitating computations in algebraic topology.^[4] A key feature is that continuous paths in \Map(X, Y) precisely correspond to homotopies between maps X \to Y. Specifically, a path \gamma: I \to \Map(X, Y), where I = [0,1] is the unit interval, defines a homotopy H: X \times I \to Y via the evaluation map ev: \Map(X, Y) \times X \to Y, which is continuous under the compact-open topology when X is locally compact. This correspondence underpins the topological enrichment of the homotopy category, allowing homotopies to be treated as morphisms in a topological sense.^[4] The compact-open topology plays a crucial role in defining loop spaces, where the based loop space \Omega Y at a basepoint y_0 \in Y is the subspace of \Map(I, Y) consisting of maps sending $0,1 \in I to y_0, topologized via the compact-open structure on \Map(I, Y). This makes \Omega Y a topological space whose homotopy groups shift those of Y, i.e., \pi_n(\Omega Y, \gamma) \cong \pi_{n+1}(Y, y_0) for a loop \gamma, enabling iterative constructions like higher loop spaces \Omega^n Y. Similarly, in classifying spaces, the mapping space \Map(X, BG) for a topological group G with classifying space BG models the homotopy type of the space of principal G-bundles over X, with connected components corresponding to conjugacy classes of homomorphisms from \pi_1(X) to G (when G is discrete).^[4] In the category of compactly generated Hausdorff spaces, denoted \CGH, the compact-open topology endows the internal hom-object \Map(X, Y) with the structure needed for \CGH to be cartesian closed, satisfying the exponential law \Map(X, \Map(Y, Z)) \cong \Map(X \times Y, Z) naturally as homeomorphisms. This closed structure supports enriched category theory in homotopy, allowing limits and colimits to interact well with mapping spaces.^[12] The singular functor \Sing: \Top \to \sSet, which assigns to a space Y its singular simplicial set with n-simplices as maps \Delta^n \to Y, forms a right adjoint to the geometric realization functor |\cdot|: \sSet \to \Top. The unit and counit of this adjunction are natural transformations whose components are continuous maps when topological spaces are equipped with the compact-open topology on function spaces, preserving the homotopy-theoretic data across the adjunction. This ensures that weak homotopy equivalences are detected properly, with |\Sing(Y)| \simeq Y for any Y.^[4]

In Functional Analysis

In the context of Pontryagin duality, the compact-open topology equips the dual group \hat{G} of a locally compact abelian (LCA) group G with a natural structure that preserves the duality. Specifically, \hat{G} consists of all continuous group homomorphisms from G to the circle group \mathbb{T}, and the compact-open topology on \hat{G} is defined by subbasis sets of the form W(K, U) = \{\chi \in \hat{G} \mid \chi(K) \subseteq U\}, where K \subseteq G is compact and U \subseteq \mathbb{T} is open. This topology ensures that \hat{G} is itself an LCA group, enabling the biduality theorem that G is topologically isomorphic to the double dual \hat{\hat{G}}. The choice of the compact-open topology is crucial for the continuity of the duality map and the validity of the Pontryagin duality theorem, which generalizes the Fourier transform to arbitrary LCA groups.^[13]^[14] The compact-open topology also arises in the study of spaces of bounded linear operators B(X, Y) between Banach spaces X and Y, where it induces the topology of uniform convergence on compact subsets of X. This topology on B(X, Y) is Hausdorff and makes composition continuous when Y is a topological vector space, facilitating the analysis of operator convergence. Notably, when X is finite-dimensional, every bounded subset of X is relatively compact, so uniform convergence on compact subsets coincides with uniform convergence on bounded sets, which in turn aligns with the operator norm topology induced by \|T\| = \sup_{\|x\| \leq 1} \|Tx\|. This equivalence simplifies the study of finite-rank approximations and stability in finite-dimensional settings.^[15]^[16] On the dual space X^* of a Banach space X, viewed as the space of continuous linear functionals L(X, \mathbb{R}) (or \mathbb{C}), the compact-open topology manifests as uniform convergence on compact subsets of X, which is strictly finer than the weak* topology of pointwise convergence on elements of X. The weak* topology, generated by seminorms p_x(\phi) = |\phi(x)| for x \in X, makes the evaluation maps X^* \to \mathbb{R} continuous, whereas the compact-open topology strengthens this to ensure continuity of functionals under uniform limits on compacts. This relationship underpins reflexivity criteria, as Mackey-Arens spaces often involve comparing these topologies on duals, and it aids in proving density results for separable preduals in the weak* topology.^[17]^[18] The Arzelà–Ascoli theorem exemplifies the compact-open topology's role in compactness for function spaces in analysis. For a compact Hausdorff space K and the space C(K, \mathbb{R}) of continuous real-valued functions on K equipped with the compact-open topology (which coincides with the uniform topology since K is compact), a subset \mathcal{F} \subseteq C(K, \mathbb{R}) is relatively compact if and only if it is pointwise relatively compact (i.e., \mathcal{F}(x) is relatively compact in \mathbb{R} for each x \in K) and equicontinuous (i.e., for every \varepsilon > 0, there exists \delta > 0 such that if d(x,y) < \delta then |f(x) - f(y)| < \varepsilon for all f \in \mathcal{F}). Equicontinuity here equates the compact-open topology with the topology of uniform convergence on K, enabling sequential compactness and applications to existence of solutions in boundary value problems.^[10] Applications extend to integral operators and approximation theory, where the compact-open topology on spaces of operators ensures that families of integral operators with equicontinuous kernels approximate solutions to integral equations on unbounded domains. For instance, collectively compact approximations—where images of the unit ball under a family of operators have compact closure in the compact-open topology—guarantee convergence of Galerkin methods for Fredholm integral equations, even when the domain is non-compact, by leveraging Ascoli-type compactness in the range space. This framework supports error estimates in numerical schemes for approximating operator inverses and resolvents in Hilbert or Banach spaces.^[19]^[20]

Variants

For Differentiable Functions

The compact-open topology extends naturally to the space C^k(M, N) of C^k-differentiable maps between smooth manifolds M and N via the k-jet prolongation j^k: C^k(M, N) \to C(M, J^k(M, N)), where J^k(M, N) denotes the bundle of k-jets over M \times N, and the codomain carries the standard compact-open topology.^[21] The resulting subbasis on C^k(M, N) consists of sets \{f \in C^k(M, N) \mid j^k(f)(K) \subseteq U\}, with K \subset M compact and U open in J^k(M, N), capturing uniform control on derivatives up to order k over compacts.^[21] For the subspace \mathrm{Diff}^k(M, N) of k-diffeomorphisms, this restricts to an analogous topology, often denoted \mathcal{W}O^k.^[21] When M is compact, this topology equips C^k(M, N) with a Fréchet manifold structure, locally modeled on Fréchet spaces of compactly supported sections of pullback bundles like \Gamma_c(f^* TN).^[22] Composition C^k(M, N) \times C^k(N, P) \to C^k(M, P) is C^\infty-continuous in this topology, preserving the chain rule for derivatives under the induced smooth structure.^[21]^[22] A sequence \{f_n\} \subset C^k(M, N) converges to f if, for every compact K \subset M, the maps and their derivatives up to order k converge uniformly on K.^[21] This framework applies to infinite-dimensional manifolds by endowing mapping spaces with smooth structures, enabling differential-geometric tools like tangent bundles and Riemannian metrics on them.^[21] In transversality theory, the Baire property of \mathrm{Diff}^\infty(M, N) under the smooth compact-open topology ensures that sets of diffeomorphisms transverse to given submanifolds are dense and open, as in Thom's transversality theorem for jets.^[21] In contrast to Sobolev topologies on differentiable maps, which impose global L^p-integrability on derivatives, the compact-open variant emphasizes uniform bounds on compacts for local analytic control.^[23]

Generalizations and Comparisons

The box topology on the space of all functions Y^X from a topological space X to Y is generated by subbasic open sets of the form \{f \in Y^X \mid f(x_i) \in U_i \text{ for } i=1,\dots,n\}, where x_1,\dots,x_n \in X and U_i are open in Y, leading to convergence that requires uniform control on finite pointwise restrictions without regard to the global structure of X. In contrast, the compact-open topology requires uniform convergence on compact subsets of X, making it strictly coarser than the box topology unless X is finite, as the box topology imposes stricter local uniformity at every finite collection of points, while the compact-open topology leverages the compactness condition to relax this on non-compact domains.^[24] The uniform topology on Y^X, generated by sets \{f \in Y^X \mid \sup_{x \in X} d(f(x), g(x)) < \epsilon\} when Y is uniformizable, is finer than the compact-open topology whenever X is not compact, since uniform convergence on the entire X implies uniform convergence on every compact subset, but the converse fails on unbounded domains like \mathbb{R}, where sequences converging compact-uniformly may diverge globally. This refinement ensures better control for applications requiring global boundedness, though it sacrifices some categorical niceties preserved by the compact-open topology.^[25] For spaces Y that are not necessarily Hausdorff, the C-compact-open topology on the set C(X) of real-valued continuous functions on a Tychonoff space X extends the standard compact-open by using closed subsets of Y in the subbasis, ensuring continuity on compacta while accommodating non-separated codomains; it coincides with the compact-open when Y is Hausdorff but provides a coarser structure otherwise, preserving topological properties like complete regularity under milder assumptions.^[26] Generalizations of the compact-open topology appear in the context of locales, where it is defined using completely prime filters to capture "compact" opens in pointless topology, yielding a cartesian closed structure on the category of locales that mirrors the exponential law for spaces. In enriched category theory over topological monoidal categories, the compact-open topology equips hom-objects with a structure that makes the category symmetric monoidal closed, extending the enrichment to settings like metric spaces viewed as [0,\infty]-categories. For non-compact X, the weak compact-open topology, generated by uniform convergence on precompact subsets, weakens the standard version to handle spaces without sufficient compacts, such as hemicompact domains, while maintaining sequential convergence properties akin to the original.^[27] The compact-open topology is always finer than the product topology on Y^X, which induces pointwise convergence, as uniform convergence on singletons (compacts) implies pointwise but not conversely on infinite X. They coincide precisely when X is finite, since all subsets are compact, reducing the compact-open subbasis to the pointwise one; for example, on X with two points, both topologies yield the same discrete structure on finite Y.^[24]

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