CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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$R / Z$ is not simply connected

Statement

$R / Z$ is not simply connected.

Facts & Assumptions

Given: The quotient circle with basepoint $[0]$ and its standard loop $ω_{1}$ .

[L1]

$R / Z$ is compact and path-connected ( $R / Z$ is compact and path-connected).

[L2]

A based circle loop is nullhomotopic exactly when its degree is zero (A based circle loop is nullhomotopic exactly when its degree is zero).

[L3]

$\deg (ω_{n}) = n$ for every integer $n$ ( $\deg (ω_{n}) = n$ for every integer $n$ ).

[L4]

A space is simply connected when it is nonempty and path-connected and its fundamental group has exactly one element at every basepoint (Simply connected topological spaces).

[L5]

The quotient circle contains its basepoint $[0]$ (The circle as $S^{1} = R / Z$ with basepoint $[0]$ ).

Proof

technique · direct

1.1

The quotient is nonempty because it contains $[0]$ by [L5], and it is path-connected by [L1].

L1L5

1.2

By [L3], $\deg (ω_{1}) = 1 \neq 0$ . The criterion [L2] therefore shows that $ω_{1}$ is not nullhomotopic, so its loop class differs from the constant-loop class.

L2L3algebra

2.1

Thus the fundamental group at $[0]$ does not have exactly one element. Although step 1.1 supplies the other two clauses of [L4], this failure at one basepoint violates the definition, so $R / Z$ is not simply connected.

step 1.1step 1.2L4∎

Depends on

Used by

A covering quotient of a simply connected space need not be simply connected Example

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 108 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

Allen Hatcher, Algebraic Topology, Ch. 1, Section 1.1, Theorem 1.7 (standard reference, not scraped)

R/ZR/Z is not simply connected