LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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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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Finite harnack chain on a compact connected subset

Statement

Let $n \geq 2$ , let $Ω \subseteq R^{n}$ be a domain, and let $K \subset Ω$ be compact, possibly empty or disconnected. There is a finite nonempty family ${B_{r_{j}} (a_{j})}_{j = 1}^{N}$ covering $K$ , with $r_{j} > 0$ and ${\overline{B}}_{4 r_{j}} (a_{j}) \subset Ω$ , whose overlap graph is connected. An edge means that the two open balls intersect. The family depends only on $K$ and $Ω$ .

Facts & Assumptions

Given: The objects and hypotheses in the statement.

[F1]

A nonempty clopen subset of a connected space is the whole space, since its nonempty complement would give a separation. (Separation of a topological space, connected and disconnected spaces, clopen sets, and connected subsets).

[F2]

Closed bounded Euclidean subsets are compact. (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).

Proof

technique · direct

1.1

Call a ball admissible if its radius is positive and its fourfold closed ball lies in $Ω$ . Every point centers such a ball by openness. Fix one admissible base ball, using nonemptiness of $Ω$ . Let $E$ be the union of all admissible balls joined to it by a finite sequence of overlapping admissible balls.

given

2.1

The set $E$ is nonempty and open. If $x \in Ω$ is a relative closure point of $E$ , an admissible ball centered at $x$ meets $E$ , hence meets a ball in one of the finite sequences. Appending the new ball proves it is in the reachable family and $x \in E$ . Thus $E$ is relatively closed, and connectedness implies $E = Ω$ .

F1step 1.1

3.1

Compactness gives a finite subcover of $K$ by admissible balls. Each selected ball meets a reachable ball by the preceding conclusion, and can be appended to a finite sequence from the base. Take the union of these finitely many finite sequences together with the base ball. The resulting finite family covers $K$ and its overlap graph is connected. If $K$ is empty, take just the base ball. All its fourfold closed balls are compact by Heine–Borel and remain in $Ω$ .

F2step 2.1given∎

Depends on

Used by

Harnack inequality on compact subsets Corollary

Dependency tree · two levels

36 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

John K. Hunter, Notes on Partial Differential Equations (2014) (standard reference, not scraped)
Tsogtgerel Gantumur, Harmonic functions (2012) (standard reference, not scraped)