DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

How 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.

The standard circle loops $ω_{n} (t) = [n t]$ for $n \in Z$

Definition

Let $I = [0, 1]$ . For every integer $n$ , define ${\tilde{ω}}_{n} (t) = n t$ and $ω_{n} = p \circ {\tilde{ω}}_{n}$ .

The function $t \mapsto n t$ is continuous by Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function, and $p$ is continuous because it is the quotient projection of The circle as $S^{1} = R / Z$ with basepoint $[0]$ . Hence $ω_{n}$ is continuous by Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous. Moreover,

$ω_{n} (0) = [0], ω_{n} (1) = [n] = [0] .$

Thus $ω_{n}$ is a based loop at $[0]$ in the sense of Based loops and the fundamental group. This definition includes $n = 0$ , when the loop is constant, and all negative integers.

The standard circle loops $ω_{n} (t) = [n t]$ for $n \in Z$

Definition

Depends on

Dependency tree · 2 levels

Sources

The standard circle loops ωn(t)=[nt]ωn​(t)=[nt] for n∈Zn∈Z

Definition

Depends on

Dependency tree · 2 levels

Sources

The standard circle loops $ω_{n} (t) = [n t]$ for $n \in Z$