Alphabeta MathSimplicial Subdivision and Simplicial Approximation: Examples
About this page
Pipeline-generated
How 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.
1 · Prerequisites
- Abelian Categories
- Categories, Functors and Natural Transformations
- Chain Complexes and Homology
- Chain Homotopy and the Homotopy Category
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Homotopy and Homotopy Equivalence
- Limits and Colimits
- Metric Spaces
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinals, Cardinals, and Transfinite Recursion
- Preadditive and Additive Categories and Biproducts
- Relations, Functions, and Quotients
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Simplicial Complexes and Simplicial Homology
- Simplicial Subdivision and Simplicial Approximation
- Subspaces, Products, and Quotients
- Suprema and Infima
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Universal Properties, Representables and the Yoneda Lemma
2 · Logical flowchart
3 · Items
Example
Barycentric subdivision of an edge and triangle
Example
The subdivision chain of an oriented two simplex
Example
A relative simplicial approximation fixed on the endpoints
Example
A compact set in an infinite non locally finite realization still meets finitely many cells
Counterexample
A continuous map need not be simplicial before subdivision
Counterexample