DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-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.

Cantor sequence space

Definition

Work in ZF. Write $N = N^{N}$ for Baire sequence space $N^{N}$ and its cylinder topology. Cantor sequence space is the subspace

$C = {x \in N : (\forall n \in N) x (n) \in {0, 1}} .$

For $s \in {0, 1}^{n}$ , let $N_{s}^{C} = {x \in C : x ↾ n = s}$ . Its topology consists of unions of these cylinders. Indeed $N_{s}^{C} = N_{s}^{N} \cap C$ , while a Baire cylinder with a nonbinary coordinate has empty intersection with $C$ . Thus this is exactly the inherited cylinder topology. The cylinder of the empty word is $C$ . The constant-zero function is a point, so no product-nonemptiness axiom is used. The constants zero and one are distinct points.

Depends on

Baire sequence space $\mathbb N^{\mathbb N}$ and its cylinder topology

Used by

The Banach–Mazur category game on sequence spaces and the real line Definition
The rational measure game Definition
Cantor and Baire sequence spaces and coordinate codings Lemma

Dependency tree · two levels

4 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

Definition 1.7 (standard reference, not scraped)