TheoremStatement: Literature-sourcedProof: 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 ultrafilter lemma, dependent choice, and a supplied family of SAFT initial objects, compact Hausdorff spaces form a reflective full subcategory of topological spaces

Statement

Assume the ultrafilter lemma and dependent choice, and suppose that an initial object of $(X ↓ I)$ is supplied for every topological space $X$ , where $I : C o m p H a u s ↪ T o p$ is the full inclusion. Then $C o m p H a u s$ is a reflective full subcategory of $T o p$ .

The supplied family is essential data and is not a consequence of the objectwise existence: under the library's convention, separate existence for each $X$ does not choose one initial object over the proper class of all topological spaces.

Facts & Assumptions

Given: The ultrafilter lemma, dependent choice, and a supplied family of initial objects of $(X ↓ I)$ , one for every topological space $X$ .

[L1]

Under these choice principles, if the objectwise SAFT initial comma objects are supplied for all topological spaces, they assemble into a left adjoint $B : T o p \to C o m p H a u s$ to the full inclusion (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).

[L2]

A full subcategory is reflective when its inclusion has a left adjoint (Reflective full subcategory and reflector).

Proof

technique · direct

1.1

The family of initial comma objects assumed in the Given is exactly the supplied family that [L1] requires, so [L1] yields the adjunction $B ⊣ I$ , whose right adjoint is the compact-Hausdorff inclusion.

L1given

2.1

Therefore [L2] says precisely that $C o m p H a u s$ is a reflective full subcategory of $T o p$ .

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 35 results over 8 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

S. Mac Lane, Categories for the Working Mathematician, compact-Hausdorff example after V.8 (standard reference, not scraped)