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Compact convergence

Compact convergence, also known as uniform convergence on compact sets, is a mode of convergence for a sequence of functions f_n from a topological space X to a metric space Y that requires the sequence to converge uniformly to a limit function f on every compact subset C \subseteq X.^[1] This notion generalizes uniform convergence by relaxing the requirement to the entire domain, making it particularly useful in spaces that are not compact, such as locally compact Hausdorff spaces.^[2] The topology of compact convergence on the space of all functions Y^X is generated by a subbasis consisting of sets of the form \{g \in Y^X \mid \sup_{x \in C} d(f(x), g(x)) < \epsilon\}, where C \subseteq X is compact, f \in Y^X, d is the metric on Y, and \epsilon > 0.^[1] In this topology, a net (or sequence) of functions converges if and only if it converges uniformly on every compact subset of X.^[2] For the subspace of continuous functions C(X, Y), this topology coincides with the compact-open topology when Y is a uniform space.^[1] Compact convergence sits between pointwise convergence and uniform convergence in strength: it is weaker than uniform convergence on the whole space but stronger than pointwise convergence, with the three coinciding under specific conditions, such as when X is compact (for uniform and compact) or discrete (for compact and pointwise).^[1] It plays a key role in functional analysis and topology, ensuring that limits of continuous functions remain continuous and preserving properties like equicontinuity on compacta.^[2] In locally compact spaces, the evaluation map from X \times C(X, Y) to Y is continuous with respect to this topology, facilitating the study of function spaces.^[1]

Definition

For Sequences of Functions

In a topological space (X, \mathcal{T}), a subset K \subseteq X is compact if every collection of open sets in \mathcal{T} that covers K admits a finite subcollection that also covers K.^[3] This property ensures that compact subsets are "small" in a topological sense, allowing control over behavior restricted to them. Compactness generalizes the Heine-Borel theorem from Euclidean spaces, where closed and bounded sets are compact, but holds more broadly without metric assumptions on X. Given a metric space (Y, d_Y), a sequence of functions f_n: K \to Y, where K is a subset of some domain, converges uniformly to a limit function f: K \to Y on K if the supremum distance between f_n and f over K approaches zero as n increases. Mathematically, this is expressed as

\lim_{n \to \infty} \sup_{x \in K} d_Y(f_n(x), f(x)) = 0.

Equivalently, for every \epsilon > 0, there exists N \in \mathbb{N} such that for all n > N and all x \in K, d_Y(f_n(x), f(x)) < \epsilon.^[4] Uniform convergence strengthens pointwise convergence by requiring the rate of approximation to be independent of the point x \in K, which is crucial for preserving properties like continuity when restricted to such sets. A sequence of functions (f_n)_{n \in \mathbb{N}}: X \to Y from a topological space (X, \mathcal{T}) to a metric space (Y, d_Y) converges compactly to a function f: X \to Y if, for every compact subset K \subseteq X, the restricted sequence (f_n|_K) converges uniformly to f|_K on K.^[5] This means that

\lim_{n \to \infty} \sup_{x \in K} d_Y(f_n(x), f(x)) = 0

holds for each such compact K.^[6] The definition requires no continuity assumptions on the functions f_n or f, though continuity is often imposed in applications to function spaces. This mode of convergence is the sequential characterization of convergence in the topology of compact convergence on the space of maps from X to Y.

For Nets of Functions

In the context of nets of functions, compact convergence generalizes the notion to arbitrary directed sets, providing a framework suitable for topological spaces where sequential convergence may not capture all limit behaviors. Consider a net (f_\alpha)_{\alpha \in A} in the function space Y^X, where X and Y are topological spaces, A is a directed set, and the codomain Y is equipped with a uniformity. The net converges compactly to a function f \in Y^X if, for every compact subset K \subseteq X, the restriction f_\alpha|_K converges uniformly to f|_K in the uniform structure on Y^K.^[7] Uniform convergence of the net on a compact set K means that \lim_{\alpha \to A} \sup_{x \in K} d_Y(f_\alpha(x), f(x)) = 0, where d_Y is a compatible metric on Y (if applicable) or more generally, the net eventually lies in entourages of the uniformity restricted to K. This limit is taken in the directed set sense: for every entourage V of the uniformity on Y, there exists \alpha_0 \in A such that for all \alpha \geq \alpha_0, (f_\alpha(x), f(x)) \in V for every x \in K. This definition ensures that compact convergence respects the local uniformity induced by compacta, forming the basis for the topology of compact convergence on Y^X.^[7] Nets are essential for defining compact convergence in general topological settings because sequences, which suffice in first-countable spaces like metric spaces, fail to describe convergence adequately in non-first-countable spaces. For instance, in spaces lacking a countable local basis at points, nets indexed by more general directed sets are required to characterize limits and continuity properly. As a special case, when the directed set A is the natural numbers with the usual order, compact convergence of nets reduces to the sequential version.^[7] Compact convergence of nets is equivalent to convergence along associated filters, particularly ultrafilters, which provide an alternative characterization without delving into the full machinery of filter convergence. This equivalence underscores the duality between nets and filters in topological convergence, allowing flexible formulations in uniform and topological contexts.^[7]

Topological Framework

Compact-Open Topology

The compact-open topology is defined on the set C(X, Y) of all continuous functions from a topological space X to a topological space Y, with subbasis consisting of the sets \{ f \in C(X, Y) \mid f(K) \subseteq V \}, where K \subseteq X is compact and V \subseteq Y is open.^[8] This topology was introduced by Richard Arens in 1946 as a structure on spaces of transformations that generalizes both the topology of pointwise convergence and the topology of uniform convergence.^[8] A net in C(X, Y) converges in the compact-open topology to a limit function if and only if it converges compactly to that function, meaning uniformly on every compact subset of X.^[9] The compact-open topology is well-defined for arbitrary topological spaces X and Y, but when X is locally compact Hausdorff, it exhibits particularly nice properties, such as making the evaluation map continuous.^[10]

Basis and Subbasis Elements

The subbasis for the compact-open topology on the function space Y^X, consisting of all functions from a topological space X to another topological space Y, is given by the collection of all sets of the form \langle K, V \rangle = \{ f \in Y^X \mid f(K) \subseteq V \}, where K is a compact subset of X and V is an open subset of Y.^[11] This subbasis generates the topology by taking all unions of finite intersections of its elements, ensuring that the resulting open sets capture uniform behavior of functions restricted to compact subsets of the domain.^[11] When restricted to the subspace C(X,Y) of continuous functions, the same subbasis induces the compact-open topology on this set.^[11] Basis elements for the compact-open topology are precisely the finite intersections of subbasis elements, that is, sets of the form \bigcap_{i=1}^n \langle K_i, V_i \rangle, where each K_i is compact in X and each V_i is open in Y.^[12] These finite intersections form a basis because the subbasis covers the entire space Y^X: for any function f \in Y^X and any compact K \subseteq X, f belongs to \langle K, Y \rangle, as Y is open in itself.^[11] For a fixed function f \in Y^X, the basic open neighborhoods in the compact-open topology are those of the form \bigcap_{i=1}^n \langle K_i, U_i(f) \rangle, where the K_i are compact subsets of X and each U_i(f) is an open neighborhood of the image f(K_i) in Y.^[12] Such neighborhoods localize the behavior of functions near f by controlling their values uniformly on the specified compact sets K_i. In the special case where Y is a metric space with metric d, the basic neighborhoods can be described using \epsilon-balls uniform on compact sets: a neighborhood of f consists of functions g such that \sup_{x \in K_i} d(f(x), g(x)) < \epsilon_i for finitely many compact sets K_i \subseteq X and \epsilon_i > 0.^[13] This formulation aligns the compact-open topology with the topology of uniform convergence restricted to compact subsets of X.^[14] The subbasis exhibits key properties that underpin the structure of the compact-open topology. It covers Y^X completely, as noted earlier, and separates points provided X and Y are Hausdorff spaces: if f \neq g, there exists x \in X with f(x) \neq g(x), and since Y is Hausdorff, disjoint open sets V_1 \ni f(x) and V_2 \ni g(x) exist such that f \in \langle \{x\}, V_1 \rangle but g \notin \langle \{x\}, V_1 \rangle, with the singleton \{x\} being compact.^[14] Under these conditions, the resulting topology is Hausdorff.^[14]

Comparisons

With Uniform Convergence

Uniform convergence of a sequence of functions \{f_n\} to f in the space of functions from a topological space X to a metric space (Y, d) is defined by the condition that \lim_{n \to \infty} \sup_{x \in X} d(f_n(x), f(x)) = 0, meaning the supremum distance over the entire domain X approaches zero.^[15] This form of convergence implies compact convergence, as uniform control over all of X restricts to uniform control on any compact subset K \subseteq X, where \sup_{x \in K} d(f_n(x), f(x)) \leq \sup_{x \in X} d(f_n(x), f(x)) \to 0.^[15] The converse holds if and only if X is compact: in this case, X itself is the sole compact subset, so uniform convergence on X is equivalent to uniform convergence on the compact set X, making the topologies coincide.^[16]^[15] When X is non-compact, compact convergence is strictly weaker than uniform convergence, permitting sequences where the supremum distance remains bounded on every compact subset but grows unbounded outside those subsets, leading to non-uniform behavior globally.^[15]

With Pointwise Convergence

Pointwise convergence of a sequence of functions \{f_n\} from a topological space X to a metric space (Y, d) to a limit function f means that for every x \in X, \lim_{n \to \infty} d(f_n(x), f(x)) = 0.^[1] This notion induces the topology of pointwise convergence on the space of all functions Y^X, which is coarser than the topology of compact convergence.^[1] Compact convergence implies pointwise convergence, since every singleton \{x\} is a compact subset of X, and uniform convergence on a singleton reduces to pointwise convergence at that point.^[1] The converse does not hold in general: pointwise convergence fails to ensure the uniformity required on larger compact sets. Uniform convergence, which is stronger than both, demands global uniformity across all of X.^[1] In the context of locally compact spaces, compact convergence is equivalent to local uniform convergence, meaning the sequence converges uniformly on every relatively compact open subset of X.^[17] This equivalence arises because locally compact spaces allow compact neighborhoods around each point, bridging the local and compact uniformities. Compact convergence demands uniformity on increasingly large compact subsets, which prevents pathological behaviors such as "escape to infinity," where the rate of convergence deteriorates as points in X move toward the boundary or infinity, even if pointwise limits hold everywhere.^[17] This stricter control distinguishes it from mere pointwise convergence, ensuring better preservation of functional properties in non-compact domains.^[1]

Properties

Preservation of Continuity

A fundamental property of compact convergence is its preservation of continuity for limits of continuous functions, provided the domain satisfies suitable topological conditions. Consider a net of continuous functions \{f_\alpha : X \to Y\}_{\alpha \in A} from a topological space X to a Hausdorff space Y, converging compactly to a function f : X \to Y. If X is a compactly generated space (also known as a k-space), then f is continuous.^[18]^[19] The proof relies on the definition of compact convergence, which ensures uniform convergence on every compact subset K \subseteq X. Thus, for each such K, the restrictions f_\alpha|_K converge uniformly to f|_K. Uniform limits of continuous functions are continuous, so f|_K is continuous on K. In a k-space, continuity on all compact subsets implies global continuity on X.^[19] If X is not compactly generated, this preservation fails in general, as counterexamples exist where the limit is continuous on compacts but discontinuous overall.^[18] Compact convergence also preserves uniform continuity on compact subsets. On any compact K \subseteq X, each f_\alpha is uniformly continuous, and the uniform limit f|_K inherits this property. However, global uniform continuity is not preserved unless X itself is compact, as the convergence need not be uniform on the entire space.^[19] In the space of all real-valued functions on X endowed with the compact-open topology—which coincides with the topology of compact convergence—the subspace C(X, \mathbb{R}) of continuous functions is closed. This closedness follows directly from the continuity preservation theorem, ensuring that limits of continuous functions remain within the subspace when X is compactly generated.^[18]

Completeness in Metric Spaces

In the context of metric spaces, the compact-open topology on the space C(X, Y) of continuous functions from a locally compact Hausdorff space X to a complete metric space Y induces a complete uniform structure. This means that every Cauchy filter in this uniform space converges to a continuous function in C(X, Y). The result holds because the entourages defined by uniform neighborhoods on compact subsets of X allow for the limit of a Cauchy sequence to be constructed pointwise on each compact set and then extended continuously using the local compactness of X. For the case where Y = \mathbb{R}, this completeness was established by R. C. Buck in his work on bounded continuous functions,^[20] with the general metric case following similarly via the metric properties of Y.^[21] Regarding metrizability, the compact-open topology on C(X, Y) is metrizable if X is compact, in which case it coincides with the topology of uniform convergence and admits a metric such as the sup metric when Y is metric. More generally, metrizability occurs when X is hemicompact (i.e., the family of compact subsets has a countable cofinal subfamily), a condition equivalent to X being \sigma-compact for locally compact Hausdorff spaces; in such cases, the topology is first-countable, and a compatible metric can be constructed using a countable exhaustion by compacts. If Y is locally compact, the topology retains similar metrizability properties under these conditions on X, but without hemicompactness, the space is not necessarily first-countable.^[22] Compactness properties within this framework are captured by the Arzelà–Ascoli theorem, which states that for compact X, any closed and equicontinuous subset of C(X, Y) (with Y metric) is compact in the compact-open topology. This provides a criterion for relative compactness in function spaces, relying on uniform boundedness and equicontinuity to ensure sequential compactness. The theorem originated from the works of Arzelà and Ascoli in the late 19th century.^[23] For \sigma-compact X, the compact-open topology on C(X, Y) admits a countable basis when Y is second-countable, facilitating sequential convergence and making the space metrizable and first-countable. This follows from the countable collection of compact subsets forming an exhaustion of X, allowing a countable subbasis of neighborhoods generated by these sets and open balls in Y. Such conditions ensure the topology behaves well for analytic purposes, like in approximation theory.^[22]

Examples

Basic Illustrations

A fundamental illustration of compact convergence is the sequence of functions f_n(x) = x^n on the open interval X = (0,1), which converges pointwise to the zero function f(x) = 0. Although this convergence is not uniform on the entire domain—since \sup_{x \in (0,1)} |f_n(x) - f(x)| = 1 for all n—it is uniform on every compact subset K \subset (0,1). For instance, on a closed subinterval K = [\delta, 1-\delta] with $0 < \delta < 1/2, the supremum norm is \|f_n - f\|_K = (1-\delta)^n \to 0 as n \to \infty, confirming compact convergence.^[24] Another straightforward example occurs with constant functions on any topological space X. Consider the sequence f_n(x) = c for all n \in \mathbb{N} and fixed c, which trivially converges compactly to the constant function f(x) = c. On any compact subset K \subset X, the uniform norm \sup_{x \in K} |f_n(x) - f(x)| = 0 for all n, so the convergence is immediate and uniform everywhere. This highlights how compact convergence encompasses uniform convergence when the domain is compact but extends naturally to non-compact settings. A classic counterexample demonstrating that pointwise convergence does not imply compact convergence is the sequence of functions f_n(x) = x^n on the space X = (0,1], where the pointwise limit is f(x) = 0 for x \in (0,1) and f(1) = 1. Although the convergence is pointwise, it fails to be uniform on the compact subset K = [1/2,1], as \sup_{x \in K} |f_n(x) - f(x)| = 1 for all n, since \sup_{x \in [1/2,1)} x^n = 1 (approached as x \to 1^-) while f(x) = 0 on [1/2,1). Similarly, on compacts like [1/n,1], the supremum of f_n is 1 and does not tend to 0.^[25] In non-locally compact spaces, such as the space of irrational numbers with the subspace topology from \mathbb{R}, compact convergence of continuous functions does not necessarily preserve continuity. For instance, in a countable metric fan M = (\omega \times \omega) \cup \{\infty\}, which is not locally compact at \infty, the compact-open topology on C_k(M, 2) contains closed copies of the Arens space S_2, showing that limits of continuous functions in this topology can fail to be continuous. This contrasts with locally compact Hausdorff spaces, where such limits are continuous.^[26] The compact-open topology is not metrizable when the domain X is not σ-compact. For example, if X is an uncountable discrete space, the compact subsets are finite, making the compact-open topology equivalent to the topology of pointwise convergence on Y^X, which has character equal to the cardinality of X and thus is not first-countable or metrizable. In such cases, sequences alone may not suffice to characterize convergence, requiring nets or filters. The Arzelà–Ascoli theorem highlights the role of equicontinuity in compact convergence: the sequence f_n(x) = \sin(n x) on [0,1] is bounded but not equicontinuous, as for any \delta > 0 and x_0 = 0, there exists n such that |\sin(n x) - \sin(0)| = 1 for some x with |x| < \delta. Consequently, no subsequence converges compactly (i.e., uniformly, since [0,1] is compact).^[27]

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