Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-17
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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.
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Degree defines a function Deg⁡:π1(S1,[0])→Z

Statement

Degree defines a function Deg⁡:π1(S1,[0])→Z by

Deg⁡([α])=deg⁡(α).

Facts & Assumptions

Given: The based circle loops and path-homotopy classes defining π1(S1,[0]).

[L1]

Path-homotopic based circle loops have the same degree (Path-homotopic based circle loops have the same degree).

[L2]

The fundamental group set of X at x0 is the set of endpoint-fixed path-homotopy classes of based loops at x0 (Based loops and the fundamental group).

Proof

technique · direct
1.1L1L2

If [α]=[β] in the set of [L2], then α and β are path-homotopic relative to their endpoints, so [L1] gives deg⁡(α)=deg⁡(β).

2.1step 1.1L2∎

Therefore the displayed rule is independent of the representative and defines one function on π1(S1,[0]). No representative-selection function is used.

Depends on

Used by

Dependency tree · two levels

7 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