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
Used by
- A discrete embedded submanifold is locally closed and countable Corollary
- Connection on a smooth vector bundle Definition
- Pullback connection Definition
- Vector field and section along a smooth curve Definition
- Assuming choice, the lower-limit plane is first countable, separable, and ccc, but not second countable or Lindelöf Example
- An open subset of a smooth manifold has a canonical restricted smooth structure Proposition
- Implication, preservation, counterexample, and choice ledger for the countability axioms Remark
- The Borel product of Rᵐ and Rⁿ is the Borel sigma-algebra of Rᵐ⁺ⁿ Theorem
- Under Dependent Choice, a subspace of a Polish space is Polish exactly when it is G_δ Theorem
Dependency tree · two levels
9 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)