ExampleConstruction: AI-adaptedVerification: AI-adaptedprecheck 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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

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

1.1L1

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

1.2L2L3

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

2.1step 1.1step 1.2L4∎

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.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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, 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)