Alphabeta MathHow 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.
Regions of the complement of a planar set and their frontiers
Definition
Let , with the usual metric topology from as the set of functions , and , , are metrics on it. A region of the complement of is a connected component of the subspace (Connected components, quasicomponents, and totally disconnected spaces, Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).
The frontier of a subset is
equivalently the set of points every open ball about which meets both and its complement, as in Interior, closure, boundary, exterior, derived set and isolated point in a topological space. A region may be bounded or unbounded; these words concern the subset of the metric plane, not the combinatorial graph drawn in it.
Depends on
- Connected components, quasicomponents, and totally disconnected spaces
- Interior, closure, boundary, exterior, derived set and isolated point in a topological space
- Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace
- $\mathbb{R}^n$ as the set of functions $n \to \mathbb{R}$, and $d_1$, $d_2$, $d_\infty$ are metrics on it
Used by
- Plane embeddings of finite simple graphs, their faces, facial boundary walks and lengths (counting a bridge twice), and planar graphs Definition
- Every plane triangulation with at least three vertices is connected Lemma
- Every point off a polygon admits a ray meeting it transversely in finitely many nonvertex points Lemma
- The complement of a polygonal arc in ℝ² is polygonally connected Lemma
- The parity of transverse ray crossings with a polygon is locally constant on its complement Lemma
- A two-connected plane graph of order at least three is maximal exactly when every face is triangular Proposition
- Polygonal Jordan curve theorem: a polygon has exactly two complementary regions and is the frontier of each Theorem
Dependency tree · two levels
30 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
- R. Diestel, Graph Theory, 6th ed., Chapter 4, Section 4.1 (standard reference, not scraped)