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.
An uncountable disjoint union of points is not second-countable
Statement refuted
An arbitrary disjoint union of second-countable manifolds is second-countable.
Facts & Assumptions
Given: An uncountable disjoint union of one-point spaces.
The A-page false statement already proves that such a space is discrete and admits no countable basis (An arbitrary disjoint union of second-countable manifolds need not be second-countable).
The topology is the disjoint-union topology of The disjoint union (coproduct) with the final topology of the canonical injections: a set is open exactly when each of its traces is, and second countability means existence of an at most countable basis (Second countability: an at most countable basis for the topology).
Counterexample
By [L1], every singleton of is open and any basis of must contain [L1] uncountably many distinct singleton sets.
Therefore is not second countable in the sense of [F1].
So is the desired counterexample.
Depends on
- An arbitrary disjoint union of second-countable manifolds need not be second-countable
- The disjoint union (coproduct) $\bigsqcup_i X_i$ with the final topology of the canonical injections: a set is open exactly when each of its traces is
- Second countability: an at most countable basis for the topology
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
11 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
- Rob van der Vorst, Introduction to differentiable manifolds, §1 (standard reference, not scraped)
- Nigel Hitchin, Differentiable Manifolds, §2.3 (standard reference, not scraped)