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.
A complex domain is a nonempty connected open subset of
Definition
A complex domain is a nonempty, connected, open subset . Open means open for of The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane in the metric topology of The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement (a topology by Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed). Connected has the meaning of Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets for this topology and its subspace topology on .
Depends on
The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane
The metric topology: a set is open when every one of its points has a ball around it inside the set; closed means open complement
Arbitrary unions and finite intersections of open sets are open, open balls are open and closed balls are closed
Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets