Alphabeta Math

Function Space Topologies and the Exponential Law: Examples and Counterexamples

About this page
How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

  • Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
  • AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
  • AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

✓ 8 results · all verified · 7 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 1 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

C([0,1], ℝ) is complete, and on it the uniform metric and the supremum metric induce the same topology

Example

The moving spikes on [0,1] converge pointwise to 0, do not converge uniformly, and do not converge in the topology of compact convergence

Counterexample

Refuted: convergence uniformly on every compact subset of ℝ implies uniform convergence. The maps x ↦ x/(n+1) separate the two

Example

On C(ℝ, ℝ) the compact-open topology has the sets {g : sup_[-m,m] |f-g| < ε} as a neighbourhood base, and ℝ is locally compact so evaluation is continuous

Example

The 1-Lipschitz maps of a metric space into ℝ form a uniformly equicontinuous family, and the distance functions x ↦ d(x,A) all belong to it

Counterexample

Refuted: a pointwise bounded family of continuous functions is equicontinuous. The spikes are bounded by 1 everywhere and are not equicontinuous at 0

Counterexample

Refuted: C(X,Y) is closed in the topology of pointwise convergence. The ramps on [0,1] converge pointwise to a discontinuous limit

Example

The map (x,z) ↦ x · z on ℝ × ℝ and its transpose z ↦ (x ↦ x · z) traced through the exponential law

Example

Dini's theorem applied to a nondecreasing sequence of piecewise linear approximations on [0,1], and what fails when the limit is not continuous

Sources