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.
Under dependent choice, every compact Hausdorff space embeds in a unit cube
Statement
Assume dependent choice. Every compact Hausdorff space embeds in a cube for some set .
Facts & Assumptions
Given: Dependent choice and a compact Hausdorff space .
Under dependent choice, a compact Hausdorff space is Tychonoff (Under dependent choice a compact Hausdorff space is Tychonoff, and its disjoint closed sets are separated by continuous functions).
Proof
By [L1], is Tychonoff. The forward implication of A space is Tychonoff if and only if it embeds in a cube then supplies an embedding of into a unit cube.
This is the asserted embedding.
Depends on
- Under dependent choice a compact Hausdorff space is Tychonoff, and its disjoint closed sets are separated by continuous functions
- A space is Tychonoff if and only if it embeds in a cube $[0,1]^J$
- The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain
Used by
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 85 results over 16 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
- E. Moorhouse, The Stone–Čech Compactification (standard reference, not scraped)