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.
Second countability is hereditary
Statement
Every subspace of a second countable space is second countable.
Facts & Assumptions
Given: A countable basis of and a subspace .
Proof
The nonempty traces for form a countable basis for the subspace topology.
Hence every subspace is second countable.
Depends on
Second countability: an at most countable basis for the topology
Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
Hereditary, open-hereditary and closed-hereditary properties of topological spaces