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

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A path homotopy over a two-set open cover admits a finite subordinate grid

Statement

Let $X = U \cup V$ with $U$ and $V$ open, and let $H : I \times I \to X$ be a path homotopy. Suppose finite subdivisions of the bottom and top edges are prescribed. Then there are finite partitions

$0 = s_{0} < s_{1} < \dots < s_{m} = 1, 0 = t_{0} < t_{1} < \dots < t_{r} = 1$

such that the horizontal partition contains every prescribed bottom and top cut, and every closed grid rectangle $[s_{i - 1}, s_{i}] \times [t_{k - 1}, t_{k}]$ is mapped by $H$ wholly into $U$ or wholly into $V$ .

Facts & Assumptions

Given: The open cover, path homotopy, and the two prescribed finite boundary subdivisions in the Statement.

[F1]

Closed boxes in $R^{2}$ are compact metric spaces (Heine-Borel in $R^{n}$ : with the Euclidean metric a subset of $R^{n}$ is compact if and only if it is closed and bounded, and the proof by bisection uses no choice principle; the same holds on the real line).

[F2]

Every open cover of a compact metric space has a positive Lebesgue number (Every open cover of a compact metric space has a Lebesgue number: a $δ > 0$ such that every nonempty subset of diameter less than $δ$ lies inside a single member of the cover).

[F3]

The product topology on $R^{2}$ is the topology of the sup metric $d_{\infty}$ , whose balls are open boxes (For $n \geq 1$ the product topology on $n$ copies of the usual topology of $R$ is the metric topology of $d_{\infty}$ on $R^{n}$ , and hence also of $d_{1}$ and $d_{2}$ , so $R^{n}$ as a product and $R^{n}$ as a metric space are one space).

[F4]

A continuous map has open preimages of open sets (For a map of spaces the following agree: continuity at every point, preimages of open sets open, preimages of closed sets closed, preimages of subbasic open sets open, and $f (\overline{A}) \subseteq \overline{f (A)}$ ).

[F5]

For every real $η > 0$ there is a natural $q \geq 1$ with $1 / q < η$ (For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ ).

Proof

technique · constructive

1.1

By [F4], $H^{- 1} (U)$ and $H^{- 1} (V)$ form an open cover of the compact metric square $I^{2}$ . By [F1] and [F2], choose a Lebesgue number $δ > 0$ for this cover in $d_{\infty}$ .

F1F2F3F4

2.1

Take $q \geq 1$ with $1 / q < δ$ , using [F5]. Start with the uniform cuts $k / q$ in each coordinate, adjoin the finitely many prescribed bottom and top cuts to the horizontal list, and sort each resulting finite set after removing repetitions. Every gap in either partition is at most $1 / q < δ$ , and both lists contain $0$ and $1$ .

step 1.1F5construct

3.1

Each closed grid rectangle is nonempty and has $d_{\infty}$ -diameter at most $1 / q < δ$ , so [F2] places it inside $H^{- 1} (U)$ or $H^{- 1} (V)$ . Its image therefore lies in the corresponding cover member, and the constructed horizontal partition refines both prescribed boundary subdivisions.

step 1.1step 2.1F2discharge-construct∎

Depends on

Used by

Homotopic-loop factorizations have the same value in the group pushout Lemma

Dependency tree · two levels

64 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, proof of Theorem 1.20 (standard reference, not scraped)
J. Peter May, A Concise Course in Algebraic Topology, Chapter 2, Section 7 (standard reference, not scraped)