TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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.
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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 $I : C o m p H a u s ↪ T o p$ . If these initial objects are supplied for all $X$ , they assemble into a left adjoint $B : T o p \to C o m p H a u s$ .

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

1.1L1L2construct

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

2.1step 1.1L2

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

3.1step 2.1L3L4discharge-construct∎

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 $B ≅ β$ .

Depends on

Used by

With the ultrafilter lemma, dependent choice, and a supplied family of SAFT initial objects, compact Hausdorff spaces form a reflective full subcategory of topological spaces Theorem

Dependency tree · two levels

30 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

S. Mac Lane, Categories for the Working Mathematician, compact-Hausdorff example after V.8 (standard reference, not scraped)
E. Riehl, Category Theory in Context, example 4.7.12 (standard reference, not scraped)