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.

A countable shrinking computed

Example

In a discrete space every decreasing closed sequence $(F_{n})$ with empty intersection is its own open expansion: $G_{n} = F_{n}$ has $\overline{G_{n}} = F_{n}$ . Explicitly, on discrete $ω$ take $F_{n} = {m \in ω : m \geq n}$ and $G_{n} = F_{n}$ . No choice axiom is needed.

Facts & Assumptions

Given: A discrete space $X$ (every subset is open), and decreasing closed $F_{n} \subseteq X$ with empty intersection; in the displayed instance $X = ω$ and $F_{n} = {m : m \geq n}$ .

[F1]

Countable paracompactness asks for a locally finite open refining cover of each countable open cover (Countable paracompactness and Dowker spaces).

Verification

1.1givenalgebra

In a discrete space complements of subsets are open, so every subset is also closed. Thus $G_{n} = F_{n}$ is open, contains $F_{n}$ , and has $\overline{G_{n}} = F_{n}$ . Consequently $⋂_{n} \overline{G_{n}} = ⋂_{n} F_{n} = \emptyset$ . The same equations hold when $F_{n}$ or $X$ is empty.

2.1F1step 1.1

For the instance on $ω$ , $F_{0} = ω$ , $F_{1} = ω ∖ {0}$ , and $F_{n + 1} \subseteq F_{n}$ . For each $m \in ω$ , $m \notin F_{m + 1}$ , proving the empty intersection despite every $F_{n}$ being nonempty. The singleton family ${{m} : m \in ω}$ is an open refining cover of every open cover: a member containing $m$ also contains ${m}$ . The neighborhood ${m}$ meets exactly one singleton, so the family is locally finite, verifying countable paracompactness directly. No simultaneous selection of cover members is involved. QED.

Depends on

Countable paracompactness and Dowker spaces

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

3 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

K. P. Hart, Set-Theoretic Topology, Chapter 4 §3 Theorem 3.3(3), p. 27; discrete specialization (standard reference, not scraped)