CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-06

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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No uniform subdivision depth for all singular simplices

Statement refuted

For a fixed nontrivial two-open cover of $[0, 1]$ and integer coefficients, no single integer $r$ makes $S^{r} σ$ cover-small for every singular $1$ -simplex $σ$ taken with coefficient $1$ .

Counterexample

Given: A proposed depth $r$ and overlapping proper open intervals $U, V$ covering $[0, 1]$ .

Proof technique: direct.

1.1

Choose $a \in U ∖ V$ and $b \in V ∖ U$ . On each of the $2^{r}$ dyadic subintervals of the parameter interval, define $σ$ linearly from $a$ to $b$ or from $b$ to $a$ , alternating orientations so the pieces join continuously.

givenconstruct

2.1

Every singular $1$ -simplex occurring in $S^{r} σ$ has image containing both $a$ and $b$ , hence lies in neither $U$ nor $V$ . In degree one all these subdivision summands have coefficient $+ 1$ over $Z$ ; coincident summands add rather than cancel. Thus a non-small term survives and $S^{r} σ$ is not cover-small, disproving uniformity.

step 1.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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, Proposition 2.21 (standard reference, not scraped)