Alphabeta Math
TheoremStatement: AI-adaptedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16‡ rests on unproved material (inherited)
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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

With the SAFT initial comma objects supplied for all spaces, they assemble into the compact-Hausdorff reflection and agree on Tychonoff spaces with the constructed Stone-Cech adjunction

Statement

Assume the ultrafilter lemma and dependent choice. For every topological space X, objectwise SAFT gives an initial object of (X↓I) for the inclusion I:CompHaus↪Top. If these initial objects are supplied for all X, they assemble into a left adjoint B:Top→CompHaus.

After restricting the domain to Tychonoff spaces, B is naturally isomorphic, as a left adjoint to the same inclusion, to the chosen Stone-Cech compactification functor β.

Facts & Assumptions

Given: The ultrafilter lemma, dependent choice, and a supplied family of the objectwise initial comma objects.

[L1]

Under these choice principles, the compact-Hausdorff inclusion satisfies the explicit supplied-well-powering SAFT hypotheses (Under the ultrafilter lemma and dependent choice, compact Hausdorff spaces satisfy the explicit SAFT hypotheses for their inclusion into topological spaces).

[L2]

Objectwise SAFT gives initial comma objects, and a supplied family of them assembles into a left adjoint (Special adjoint functor theorem, objectwise form with explicit intersection smallness or preservation data, Special adjoint functor theorem, data-supplied functor form).

[L3]

On Tychonoff spaces, the chosen Stone-Cech functor is left adjoint to the compact-Hausdorff inclusion (Under the ultrafilter lemma and dependent choice, Stone-Cech compactification is left adjoint to the compact-Hausdorff inclusion).

[L4]

Two left adjoints to the same functor are naturally isomorphic by a unique natural isomorphism compatible with the adjunctions (Adjoints are unique up to a unique natural isomorphism compatible with the adjunction data).

Proof

technique · constructive
1.1

For each topological space X, [L1] and the objectwise part of [L2] give an initial object of (X↓I).

L1L2construct
2.1

Applying the functor form of [L2] to the supplied family assembles these universal arrows into B⊣I.

step 1.1L2
3.1

Restrict B and I to Tychonoff spaces. By [L3], both B and β are left adjoint to the same compact-Hausdorff inclusion, so [L4] supplies the unique compatible natural isomorphism B≅β.

step 2.1L3L4discharge-construct∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 99 results over 14 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources