LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31‡ 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.

Under choice, if $∣ I ∣ > 2^{ℵ_{0}}$ , then the Cantor cube $2^{I}$ is not separable

Statement

Assuming choice, $∣ I ∣ > 2^{ℵ_{0}}$ implies $2^{I}$ is not separable.

Facts & Assumptions

Given: Choice, $∣ I ∣ > 2^{ℵ_{0}}$ , and the product topology on $2^{I}$ .

[L1]

Every nonempty at most countable set is a surjective image of $N$ (A nonempty set is at most countable iff it is a surjective image of $N$ ).

[L2]

The binary sequences have cardinality $2^{ℵ_{0}}$ (Cardinal sum $κ \oplus λ$ , product $κ \otimes λ$ and exponentiation $κ^{λ}$ , and why they are written apart from the ordinal operations, Assuming the Axiom of Choice, $2^{κ} = ∣ P (κ) ∣$ , and Cantor's theorem in cardinal form: $κ < 2^{κ}$ ).

[F1]

A condition on finitely many coordinates defines a basic open cylinder (The product set $\prod_{i \in I} X_{i}$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space).

Proof

technique · contradiction

1.1

Suppose $D \subseteq 2^{I}$ is at most countable and dense. It is nonempty because $2^{I}$ is nonempty, so choose a surjection $s : N \to D$ by [L1]. For each $i \in I$ define its column $c_{i} \in 2^{N}$ by $c_{i} (n) = s (n) (i)$ .

L1assume-contraconstruct

2.1

By [L2] there are only $2^{ℵ_{0}}$ possible columns, whereas $∣ I ∣ > 2^{ℵ_{0}}$ . Thus distinct $i, j \in I$ have $c_{i} = c_{j}$ , which says $d (i) = d (j)$ for every $d \in D$ because $s$ is onto.

step 1.1L2

3.1

The cylinder ${x \in 2^{I} : x (i) = 0, x (j) = 1}$ is nonempty and open by [F1], but step 2.1 makes it disjoint from $D$ , contradicting density. Hence $2^{I}$ is not separable.

step 2.1F1discharge-contradiction∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 86 results over 25 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

UCR General Topology Notes (standard reference, not scraped)
Cantor cube (Wikipedia) (standard reference, not scraped)

Under choice, if ∣I∣>2ℵ0∣I∣>2ℵ0​, then the Cantor cube 2I2I is not separable