False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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.

Assuming choice, refuted: every ccc space is separable

Statement

Every ccc space is separable.

Facts & Assumptions

Given: The Axiom of Choice and an index set $I$ with $∣ I ∣ > 2^{ℵ_{0}}$ .

[L1]

Under choice every Cantor cube $2^{I}$ satisfies ccc (Under choice, every Cantor cube $2^{I}$ satisfies ccc).

[L2]

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

Refutation

technique · direct

1.1

Let $X = 2^{I}$ with its product topology.

given

1.2

The space $X$ satisfies the hypothesis of the proposed implication because it is ccc by [L1].

2.1

The same space fails the proposed conclusion because it is not separable by [L2].

step 1.1L2

3.1

Thus $X$ is a ccc nonseparable space, which refutes the statement.

step 1.2step 2.1∎

Depends on

Used by

Implication, preservation, counterexample, and choice ledger for the countability axioms Remark

Dependency tree · two levels

11 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

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