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 in the modulus metric 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, and connected has the meaning of Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets. By as the Euclidean plane and as a normed real algebra: what the identification preserves, these are exactly the usual Euclidean notions for the corresponding subset of .
Depends on
- $\mathbb C=\mathbb R[x]/(x^2+1)$ as the Euclidean plane and as a normed real algebra: what the identification preserves
- 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
- Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets
Used by
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 59 results over 12 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
- J. Lebl, Guide to Cultivating Complex Analysis, §2.1 (standard reference, not scraped)
- R. Howell and J. Mathews, Complex Analysis, §3.1 (standard reference, not scraped)