ExampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-10

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.

The Stone space of a finite powerset algebra

Example

For a finite set $S$ , the Stone space of $P (S)$ is the discrete space $S$ : its points are $U_{s} = {A \subseteq S : s \in A}$ , and the basic clopen $[A]$ corresponds to $A$ . This includes $S = \emptyset$ and uses no choice principle.

Facts & Assumptions

[F1]

Finite Boolean algebras are powersets of their atoms says ultrafilters of a finite Boolean algebra are generated by its atoms.

[F2]

Stone ultrafilter space and its clopen basis defines $[A]$ by membership of $A$ in an ultrafilter.

Verification

Given: A finite set $S$ , with the Boolean operations on $P (S)$ being intersection, union and relative complement.

1.1

The nonzero elements are the nonempty subsets. Each singleton ${s}$ is minimal nonzero; a subset with two distinct elements has a nonempty proper singleton subset, so is not an atom. Thus the atoms are exactly the singletons. F1 now lists all ultrafilters as $U_{s}$ , uniquely for $s \in S$ .

F1givenalgebra

2.1

For $A \subseteq S$ , $U_{s} \in [A]$ exactly when $A \in U_{s}$ , exactly when $s \in A$ . Therefore $[A] = {U_{s} : s \in A}$ and $[{s}] = {U_{s}}$ . Every singleton in the Stone space is open, so every subset is open. For the explicit instance $S = {0, 1}$ , the four algebra elements give respectively $[\emptyset] = \emptyset$ , $[{0}] = {U_{0}}$ , $[{1}] = {U_{1}}$ and $[S] = {U_{0}, U_{1}}$ .

F2step 1.1algebra

3.1

If $S = \emptyset$ , the algebra has one element and no atoms; F1 gives no ultrafilters, and the empty map identifies the two empty spaces. If $S$ has one point, step 2.1 gives a one-point discrete Stone space with exactly two clopens. This verifies the claimed identification in every finite case. QED.

F1step 1.1step 2.1algebra

Used by

Nothing in the library uses this result yet.

Dependency tree · 0 levels

Nothing. This result depends on no other item in the library.

Sources

Tressl, Stone Duality for Boolean Algebras, 2.3.2 and 3.1.3; finite powerset calculation (standard reference, not scraped)