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.
A compact regular level surface is covered by finitely many regular surface patches
Statement
If a regular level surface is compact, then finitely many regular surface patches have relative interiors whose union contains . The empty surface is covered by the empty family.
Facts & Assumptions
Given: A compact regular level surface .
Every point of lies in the relative interior of a regular surface patch (Regular level surfaces have local regular parametrizations with the same tangent plane).
Compactness is intrinsic to the subspace metric, and every open cover of a compact metric space has a finite subcover (A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it, Open cover, subcover, compact metric space, and compact subset of a metric space).
Proof
If , the empty family covers it. Otherwise, for each , [L1] gives a patch whose relative interior contains ; these relative interiors form an open cover of in its subspace topology.
By [L2], select a finite subcover. The corresponding finite list of regular patches covers .
Together with the empty case in step 1.1, this proves the statement for every compact regular level surface.
Depends on
- Regular level surfaces have local regular parametrizations with the same tangent plane
- Open cover, subcover, compact metric space, and compact subset of a metric space
- A subset of a metric space is open in the subspace metric exactly when it is the trace of an open set of the ambient space, and it is compact as a metric space in its own right exactly when every family of open subsets of the ambient space covering it, indexed or not, has finitely many members covering it
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
17 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
- M. E. Taylor, Introduction to Analysis in Several Variables, Section 3.2 (standard reference, not scraped)