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.
The tail-filter and derived-net constructions preserve convergence and cluster points in both directions
Statement
Passing from a net to its tail filter, or from a filter to its derived net, preserves and reflects convergence and cluster points.
Facts & Assumptions
Given: A net, a filter, and a point of the relevant topological space.
A net and its tail filter have the same limits and cluster points (A net and its tail filter have the same limits and cluster points).
A filter and its derived net have the same limits and cluster points (A filter and its canonical derived net have the same limits and cluster points).
Proof
Apply [L1] to the given net.
Apply [L2] to the given filter.
These are precisely the two asserted correspondences.
Depends on
Used by
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 12 results over 5 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
- WVU Math 581 Topology I (standard reference, not scraped)
- Net (mathematics) (Wikipedia) (standard reference, not scraped)
- Filter (set theory) (Wikipedia) (standard reference, not scraped)