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.
Simply connected topological spaces
Definition
A topological space is simply connected when it is nonempty and path-connected (Paths, path-connected spaces and path components) and, for every , the group has exactly one element.
Requiring every basepoint avoids presuming a change-of-basepoint theorem. For a path-connected space that later theorem shows that checking one basepoint is equivalent, but no such result is needed for this definition. The empty space is path-connected under the published convention, but it is not simply connected here because nonemptiness is explicit.
Depends on
Used by
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 78 results over 18 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
- A. Hatcher, Algebraic Topology, Chapter 1, Proposition 1.6 (standard reference, not scraped)