False statementConstruction: AI-adaptedVerification: 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.

Assuming choice and countable choice, refuted: arbitrary products of second countable spaces are second countable

Statement

Assuming choice and countable choice, arbitrary products of second countable spaces are second countable.

Facts & Assumptions

Given: Choice, countable choice, and an index set $I$ with $|I|>2^{\aleph_0}$ .

[L1]

The two-point discrete space is second countable, and the product topology on the family of those factors is the Cantor cube $2^I$ (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies, 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).

[L2]

The Cantor cube $2^I$ is not separable when $|I|>2^{\aleph_0}$ (Under choice, if $|I|>2^{\aleph_0}$ , then the Cantor cube $2^I$ is not separable).

[L3]

Assuming countable choice, every second countable space is separable (Assuming countable choice, every second countable space is separable).

Refutation

technique · direct

1.1

For each $i\in I$ , let $X_i=\{0,1\}$ with the discrete topology; each $X_i$ is second countable.

1.2

Their product is the Cantor cube $2^I$ by [L1].

2.1

The product $2^I$ is not separable by [L2].

step 1.2L2

3.1

If $2^I$ were second countable, [L3] would make it separable, contradicting step 2.1; hence this product of second countable spaces is not second countable.

step 2.1L3

4.1

This family of factors refutes the claimed arbitrary-product principle.

step 1.1step 3.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 69 results over 20 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)
Second-countable space (Wikipedia) (standard reference, not scraped)
Cantor cube (Wikipedia) (standard reference, not scraped)