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.
The empty space has a unique sheaf section over the empty open set
Example
Let . Then the only open set is , and every sheaf on has exactly one section over that open set.
Facts & Assumptions
Given: A sheaf on the empty space.
A topology on the underlying set has only one open subset, namely (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison).
A sheaf has a unique section over the empty open set (A set-valued sheaf has a unique section over the empty open set).
Verification
By [F1], there is no open set to consider except , so the whole presheaf data of is just the set together with its identity restriction map.
By [L1], this set is a singleton. Therefore every sheaf on the empty space has exactly one section over its only open set.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
5 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
- The Stacks Project, Sheaves on Spaces, Section 2 and Remark 7.2 (standard reference, not scraped)