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.
The Stone–Čech compactification by its compact-Hausdorff extension property
Definition
A Stone–Čech compactification of is a Hausdorff compactification (A Hausdorff compactification as a dense embedding into a compact Hausdorff space) such that for every compact Hausdorff space and continuous map (Continuity of a map of topological spaces at a point and globally), there is a unique continuous with . The universal property, rather than a particular construction, is the definition.
Depends on
Used by
- Stone–Čech compactifications are uniquely homeomorphic over the original space Corollary
- The Stone–Čech compactification of a compact Hausdorff space adds no points Corollary
- Under dependent choice and the ultrafilter lemma, the Stone-Cech compactification maps continuously onto the Samuel compactification Corollary
- The one-point compactification of discrete ℕ is not βℕ Counterexample
- Every function from discrete ℕ to [0,1] extends uniquely to βℕ Example
- Under dependent choice and the ultrafilter lemma, the Samuel compactification of the discrete natural numbers is beta N Example
- FALSE: every Hausdorff compactification has the Stone–Čech extension property False statement
- Under the ultrafilter lemma and dependent choice, the closure of the full evaluation image is the Stone–Čech compactification Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 21 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- Stacks Project, Stone–Čech compactification (standard reference, not scraped)