LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

How 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.

A net and its tail filter have the same limits and cluster points

Statement

For a net $x$ and its tail filter $F_{x}$ , a point is a limit of $x$ exactly when it is a limit of $F_{x}$ , and it is a cluster point of $x$ exactly when it is a cluster point of $F_{x}$ .

Facts & Assumptions

Given: A net $x : D \to X$ , its tail filter $F_{x}$ , and $p \in X$ .

[A1]

$A \in F_{x}$ exactly when $x$ is eventually in $A$ (The tail filter of a net).

[A2]

Net and filter convergence and cluster points have their stated neighbourhood formulations (Convergence and cluster points of a net in a topological space, Convergence and cluster points of a filter on a topological space).

Proof

technique · direct

1.1

For every neighbourhood $N$ of $p$ , $x$ is eventually in $N$ exactly when $N \in F_{x}$ by [A1]. Thus the two convergence conditions in [A2] are equivalent.

A1A2

1.2

For every neighbourhood $N$ of $p$ , $x$ is frequently in $N$ exactly when $N$ meets every tail $T_{d}$ : a point in $N \cap T_{d}$ is a value $x_{e} \in N$ with $e \geq d$ .

A1A2

2.1

If $N$ meets every tail, it meets every member of $F_{x}$ , since each such member contains a tail; conversely every tail belongs to $F_{x}$ . Hence the two cluster-point conditions in [A2] are equivalent.

step 1.2A1A2∎

Depends on

Used by

The tail-filter and derived-net constructions preserve convergence and cluster points in both directions Theorem

Dependency tree · two levels

9 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)
Filter (set theory) (Wikipedia) (standard reference, not scraped)
Net (mathematics) (Wikipedia) (standard reference, not scraped)