Function Space Topologies and the Exponential Law

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✓ 16 results · all verified · 12 also independently AI-judged

Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 4 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.

1 · Prerequisites

Abstract Algebra

Binary Operations, Monoids, Groups and Subgroups

Linear Algebra

Vector Spaces, Linear Subspaces, Span and Direct Sums

Real Analysis

Set Theory

Topology

2 · Logical flowchart

3 · Items

Definition

In a locally compact metric space every point has arbitrarily small compact closed balls, hence a neighbourhood base of compact sets

Lemma

Tube lemma: if K is a compact subset of a metric space X, Z is a topological space and N is open in X × Z with K × {z₀} ⊆ N, then K × W ⊆ N for some open W ∋ z₀

Definition

The compact-open topology on C(X,Y) for a metric domain X, with subbasis S(K,V) = {f : f[K] ⊆ V}

Lemma

For a nonempty set X and a metric space (Y,d) the uniform metric ρ̄(f,g) = supₓ min{d(f(x),g(x)), 1} is a metric on Y^X

Definition

Uniform convergence, and the topology of uniform convergence: the metric topology of the uniform metric on Y^X and on C(X,Y)

Lemma

On C(X,Y) with X and Y metric, uniform convergence is finer than compact convergence, which is finer than pointwise convergence

Theorem

If f : X × Z → Y is continuous then its transpose F : Z → C(X,Y), F(z)(x) = f(x,z), is continuous for the compact-open topology, with no hypothesis on X beyond being metric

Theorem

The exponential law: for a locally compact metric X and any spaces Z and Y, transposition is a bijection between C(X × Z, Y) and C(Z, C(X,Y)) with the compact-open topology

Definition

Equicontinuity at a point, uniform equicontinuity, and pointwise boundedness of a family of maps between metric spaces

Theorem

Dini's theorem: on a compact metric space a nondecreasing sequence of continuous real functions converging pointwise to a continuous limit converges uniformly

Remark

Standing hypotheses on this page: a metric domain, where the target must be metric, and why the compact-open topology is built from metric compactness

False statement

Function Space Topologies and the Exponential Law

3 · Items

The topology of pointwise convergence on Y^X, which is the product topology, and its restriction to C(X,Y)

A sequence converges in the topology of pointwise convergence exactly when it converges at every point

Locally compact metric space: every point has a compact neighbourhood

In a locally compact metric space every point has arbitrarily small compact closed balls, hence a neighbourhood base of compact sets

Tube lemma: if K is a compact subset of a metric space X, Z is a topological space and N is open in X × Z with K × {z₀} ⊆ N, then K × W ⊆ N for some open W ∋ z₀

The compact-open topology on C(X,Y) for a metric domain X, with subbasis S(K,V) = {f : f[K] ⊆ V}

For a nonempty set X and a metric space (Y,d) the uniform metric ρ̄(f,g) = supₓ min{d(f(x),g(x)), 1} is a metric on Y^X

Uniform convergence, and the topology of uniform convergence: the metric topology of the uniform metric on Y^X and on C(X,Y)

Convergence in the uniform metric is exactly uniform convergence: one N serving every point

The topology of compact convergence on C(X,Y) for metric X and Y: uniform convergence on each compact subset of X

For a metric domain and a metric target the compact-open topology on C(X,Y) is the topology of compact convergence

On C(X,Y) with X and Y metric, uniform convergence is finer than compact convergence, which is finer than pointwise convergence

A uniform limit of continuous functions is continuous, so C(X,Y) is closed in Y^X under the uniform metric

If (Y,d) is complete then Y^X is complete in the uniform metric, and so is C(X,Y)

The evaluation map e : C(X,Y) × X → Y, e(f,x) = f(x)

If X is a locally compact metric space then the evaluation map is continuous for the compact-open topology

If f : X × Z → Y is continuous then its transpose F : Z → C(X,Y), F(z)(x) = f(x,z), is continuous for the compact-open topology, with no hypothesis on X beyond being metric

The exponential law: for a locally compact metric X and any spaces Z and Y, transposition is a bijection between C(X × Z, Y) and C(Z, C(X,Y)) with the compact-open topology

Equicontinuity at a point, uniform equicontinuity, and pointwise boundedness of a family of maps between metric spaces

Dini's theorem: on a compact metric space a nondecreasing sequence of continuous real functions converging pointwise to a continuous limit converges uniformly

Standing hypotheses on this page: a metric domain, where the target must be metric, and why the compact-open topology is built from metric compactness

FALSE: a pointwise convergent sequence of continuous functions converges uniformly on every compact set

FALSE: the compact-open topology on C(X,Y) is metrizable for every metric X and Y

FALSE: the evaluation map on C(X,Y) with the compact-open topology is continuous for every metric X