TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6.1-sol)

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.

Compactness and discreteness are exchanged by duality

Statement

Assume the Axiom of Choice (The Axiom of Choice). Let $G$ be a locally compact Hausdorff abelian group. Then:

(1) $G$ is compact if and only if $\hat{G}$ is discrete;

(2) $G$ is discrete if and only if $\hat{G}$ is compact.

The Axiom of Choice supplies the Tychonoff-based compactness implication and the choice hypotheses of biduality used in the converses.

Facts & Assumptions

Given: A locally compact Hausdorff abelian group $G$ and the Axiom of Choice.

[F1]

If $G$ is compact then $\hat{G}$ is discrete, and if $G$ is discrete then, assuming the Axiom of Choice, $\hat{G}$ is compact. Both directions are for abelian topological groups. (Compact groups have discrete duals and discrete groups have compact duals)

[F2]

The dual of a locally compact Hausdorff abelian group is again a locally compact Hausdorff abelian group. (The dual of a locally compact abelian group is locally compact abelian, The Pontryagin dual with the compact-open topology)

[F3]

The evaluation map $Φ_{G} : G \to \hat{\hat{G}}$ is an isomorphism of topological groups; a homeomorphism carries compact subsets to compact subsets and discrete spaces to discrete spaces, and $(\hat{G}) \hat{} = \hat{\hat{G}}$ . (Pontryagin biduality: the evaluation map is a topological isomorphism, Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological, Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right, The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies)

[F4]

The assumed Axiom of Choice implies Dependent Choice, so the current biduality theorem applies with its full hypotheses. (AC implies DC implies countable choice)

Proof

1.1F1

The forward implications are exactly the two clauses of [F1]: compact $G$ gives discrete $\hat{G}$ , and discrete $G$ gives compact $\hat{G}$ .

2.1F1F2F3F4step 1.1∎

For the converses, apply [F1] to the locally compact Hausdorff abelian group $\hat{G}$ of [F2], whose dual is the bidual $\hat{\hat{G}}$ : if $\hat{G}$ is compact then $\hat{\hat{G}}$ is discrete, and $Φ_{G}$ identifies $G$ with $\hat{\hat{G}}$ by [F3], so $G$ is discrete; if $\hat{G}$ is discrete then $\hat{\hat{G}}$ is compact, so $G$ is compact. Together with step 1.1 this proves both equivalences.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

65 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Lynn H. Loomis, Introduction to Abstract Harmonic Analysis, D. Van Nostrand, 1953 (Harvard-hosted full scan) (standard reference, not scraped)
Manfred Einsiedler and Thomas Ward, Ergodic Theory with a View Towards Number Theory, Appendix C (course-hosted full text) (standard reference, not scraped)