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.
Fréchet–Urysohn spaces and sequential spaces
Definition
A topological space is Fréchet–Urysohn if, whenever , there is a sequence in converging to . Equivalently, for every , since sequential closure is always contained in closure (The sequential closure is contained in the closure, continuity implies sequential continuity, and sequential limits need not be unique).
A subset is sequentially closed if every sequence in that converges in has its limit in . The space is sequential if every sequentially closed subset is closed. Equivalently, implies for every .
Depends on
Convergence and cluster points of a sequence in a topological space, sequential continuity, and the sequential closure
The sequential closure is contained in the closure, continuity implies sequential continuity, and sequential limits need not be unique
A point lies in the closure of iff every basic neighbourhood of it meets ; the closure is the smallest closed superset and equals together with its derived set