ExampleConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge 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.

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

Example

The real line is simply connected, but its quotient by integer translations is not. The canonical projection

$p : R ⟶ R / Z$

is both a quotient map and a covering map. Thus neither a quotient map nor a covering map transfers simple connectedness from its total space to its base in general.

Facts & Assumptions

Given: The real line, its integer-translation quotient, and the canonical projection $p$ .

[L1]

If $n \geq 1$ and $C \subseteq R^{n}$ is nonempty and convex, then $C$ is simply connected (Every nonempty convex subset of $R^{n}$ is simply connected).

[L2]

$p : R \to R / Z$ is the quotient projection defining the quotient circle (The circle as $S^{1} = R / Z$ with basepoint $[0]$ ).

[L3]

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

[L4]

$R / Z$ is not simply connected ( $R / Z$ is not simply connected).

Verification

technique · direct

1.1

The real line is a nonempty convex subset of $R^{1}$ , so [L1] with $n = 1$ makes $R$ simply connected.

1.2

The same explicit map $p$ is a quotient map by [L2] and a covering map by [L3].

L2L3

2.1

Its base is not simply connected by [L4], whereas its total space is simply connected by step 1.1. Step 1.2 therefore supplies both announced failures of preservation.

step 1.1step 1.2L4∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 71 results over 17 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 (standard reference, not scraped)
J. Peter May, A Concise Course in Algebraic Topology, Ch. 1, Section 5 (standard reference, not scraped)