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.
Totally bounded uniform space
Definition
A uniform space is totally bounded if, for every entourage , there is a finite set such that . Finiteness has the library meaning of The cardinality of a finite set.
Depends on
Used by
- The Samuel compactification map need not be a uniform embedding for the original uniformity Counterexample
- Under dependent choice and the ultrafilter lemma, the Samuel compactification of the open unit interval is the closed unit interval Example
- Assuming dependent choice, a totally bounded uniformity equals its Samuel uniformity Lemma
- Every compact uniform space is totally bounded Lemma
- Every ultrafilter on a totally bounded uniform space is Cauchy Lemma
- The Samuel uniformity is totally bounded Lemma
- Total boundedness passes to a uniform space with a dense uniformly continuous image Lemma
Dependency tree · two levels
11 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
- J. Wodzicki, Uniform Structure (standard reference, not scraped)
- Encyclopedia of Mathematics, Uniform space (standard reference, not scraped)