Alphabeta MathPropositionStatement: Literature-sourcedProof: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31
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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
technique · direct
1.1given
The nonempty traces for form a countable basis for the subspace topology.
2.1step 1.1∎
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
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 21 results over 10 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)