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 Hausdorff compactification as a dense embedding into a compact Hausdorff space
Definition
A Hausdorff compactification of a space is a pair in which is compact (Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right) and Hausdorff (Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not), and is an embedding with dense image (Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological, Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets). We identify with only after naming ; the density condition is a condition on that named image.
Depends on
- Homeomorphism, open map, closed map, embedding, and what it means for a property to be topological
- Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets
- Open cover, subcover, and compact topological space; a compact subset is a subspace that is compact in its own right
- Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not
Used by
- Assuming the ultrafilter lemma, every Tychonoff space has a Hausdorff compactification Corollary
- The Samuel completion and, when compactifying, the Samuel compactification Definition
- The Stone–Čech compactification by its compact-Hausdorff extension property Definition
- FALSE: every Hausdorff compactification has the Stone–Čech extension property False statement
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 53 results over 12 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)