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

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.

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

1.1L1given

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.

2.1step 1.1L2∎

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

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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)