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.
Semilocally simply connected spaces with explicit basepoint convention
Definition
A space is semilocally simply connected at when there is a neighbourhood of and a basepoint-preserving inclusion whose induced map on fundamental groups is trivial (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open, The homomorphism on fundamental groups induced by a pointed continuous map, Based loops and the fundamental group). It is semilocally simply connected when this holds at every point. The neighbourhood need not itself be simply connected.
Depends on
Used by
- Connected coverings of the circle are classified by the subgroups nℤ for n≥0 Corollary
- The based path-class model and basic sets for a universal cover Definition
- The Hawaiian earring is locally path-connected but has no universal cover Example
- The two-circle wedge has both regular and nonregular connected three-sheeted coverings Example
- FALSE: every compact path-connected subset of ℝ² has a universal cover False statement
- For a nonempty path-connected locally path-connected semilocally simply connected space, the path-class projection is a covering map Lemma
- A space admitting a universal covering is semilocally simply connected Theorem
- Connected covering spaces are classified by conjugacy classes of fundamental-group subgroups Theorem
- Universal covering Lie group Theorem
Dependency tree · two levels
11 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
- Allen Hatcher, Algebraic Topology, §1.3 (standard reference, not scraped)
- J. Peter May, A Concise Course in Algebraic Topology, Ch. 3 (standard reference, not scraped)
- Marco Gualtieri, MAT1300 Week 4 Term 2, §1.6 (standard reference, not scraped)