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.
Assuming the ultrafilter lemma, a space is compact if and only if every universal net converges
Statement
Assume the ultrafilter lemma. A topological space is compact if and only if every universal net in it converges.
Facts & Assumptions
Given: A topological space and the ultrafilter lemma.
Compactness is equivalent to every net having a cluster point (Assuming the ultrafilter lemma, compactness is equivalent to every net having a cluster point, every net having a convergent subnet, every filter having a cluster point, and every ultrafilter converging).
Every net has a universal subnet (Assuming the ultrafilter lemma, every net has a universal subnet), and a cluster point of a universal net is a limit (Every cluster point of a universal net is a limit of that net).
A point is a cluster point of a net exactly when some subnet converges to it (A point is a cluster point of a net if and only if some subnet converges to it).
Proof
If is compact, a universal net has a cluster point by [L1], hence converges by [L2].
Conversely, suppose every universal net converges. Every net has a universal subnet by [L2], which then converges; its limit is a cluster point of the original net by [L3]. Thus every net has a cluster point.
By [L1], this makes compact.
Depends on
- Assuming the ultrafilter lemma, every net has a universal subnet
- Every cluster point of a universal net is a limit of that net
- Assuming the ultrafilter lemma, compactness is equivalent to every net having a cluster point, every net having a convergent subnet, every filter having a cluster point, and every ultrafilter converging
- A point is a cluster point of a net if and only if some subnet converges to it
Used by
Dependency tree · two levels
16 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
- WVU Math 581 Topology I (standard reference, not scraped)
- Net (mathematics) (Wikipedia) (standard reference, not scraped)
- Boolean prime ideal theorem (Wikipedia) (standard reference, not scraped)