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.
Convergence and cluster points of a net in a topological space
Definition
Let be a net in a topological space and let .
- converges to , written , if it is eventually in every neighbourhood of (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open).
- is a cluster point of if is frequently in every neighbourhood of .
Convergence implies being a cluster point. If is eventually in a neighbourhood after , then for an arbitrary threshold choose a common upper bound ; one has , so is frequently in .
Depends on
Used by
- Square-summable families on an arbitrary index set and the space ℓ²(I) Definition
- Weak convergence of nets and sequences Definition
- Weak star convergence Definition
- A neighbourhood-indexed net in A converges to each point of Ā Example
- Finite partial sums of a real family form a net directed by inclusion Example
- A filter and its canonical derived net have the same limits and cluster points Lemma
- A net and its tail filter have the same limits and cluster points Lemma
- Every cluster point of a universal net is a limit of that net Lemma
- Subnets preserve eventual properties and every limit of a net Lemma
- A map of topological spaces is continuous at a point if and only if it preserves every net converging to that point Theorem
- A point is a cluster point of a net if and only if some subnet converges to it Theorem
- A point lies in the closure of a set if and only if a net in the set converges to it Theorem
- A topological space is Hausdorff if and only if every net has at most one limit Theorem
Dependency tree · two levels
6 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)
- net (nLab) (standard reference, not scraped)