CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-21

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$R \to R / Z$ is a universal covering

Statement

The quotient projection

$p : R ⟶ R / Z, t ⟼ [t],$

is a universal covering space of the quotient circle, pointed by $0 \mapsto [0]$ .

Facts & Assumptions

Given: The quotient projection $p : R \to R / Z$ .

[F1]

The quotient projection $p$ is a covering map ( $p : R \to R / Z$ is a covering map with translated interval sheets).

[F2]

Every nonempty convex subset of Euclidean space is simply connected (Every nonempty convex subset of $R^{n}$ is simply connected).

[F3]

A universal covering is a covering map whose total space is simply connected (Universal covering spaces).

Proof

technique · direct

1.1

The map $p$ is a covering by [F1].

1.2

The real line is a nonempty convex subset of itself, so [F2] makes it simply connected.

2.1

Steps 1.1 and 1.2 satisfy both clauses of [F3], hence $p$ is a universal covering.

step 1.1step 1.2F3∎

Depends on

Used by

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, Section 1.3 (standard reference, not scraped)
J. Peter May, A Concise Course in Algebraic Topology, Chapter 3, Section 8 (standard reference, not scraped)

R→R/ZR→R/Z is a universal covering