Alphabeta MathHow 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 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 with .
The two-point discrete space is second countable, and the product topology on the family of those factors is the Cantor cube (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies, The product set 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).
The Cantor cube is not separable when (Under choice, if , then the Cantor cube is not separable).
Assuming countable choice, every second countable space is separable (Assuming countable choice, every second countable space is separable).
Refutation
For each , let with the discrete topology; each is second countable.
Their product is the Cantor cube by [L1].
The product is not separable by [L2].
If were second countable, [L3] would make it separable, contradicting step 2.1; hence this product of second countable spaces is not second countable.
This family of factors refutes the claimed arbitrary-product principle.
Depends on
- Under choice, if $|I|>2^{\aleph_0}$, then the Cantor cube $2^I$ is not separable
- Assuming countable choice, every second countable space is separable
- 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
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
27 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)
- Second-countable space (Wikipedia) (standard reference, not scraped)
- Cantor cube (Wikipedia) (standard reference, not scraped)