LemmaStatement: AI-adaptedProof: AI-generatedSession-authored (Fable 5 assisted)precheck 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 $\mathcal F_x$ , a point is a limit of $x$ exactly when it is a limit of $\mathcal F_x$ , and it is a cluster point of $x$ exactly when it is a cluster point of $\mathcal F_x$ .

Facts & Assumptions

Given: A net $x:D\to X$ , its tail filter $\mathcal F_x$ , and $p\in X$ .

[A1]

$A\in\mathcal F_x$ 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 $N\in\mathcal F_x$ 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 $e\ge d$ .

A1A2

2.1

If $N$ meets every tail, it meets every member of $\mathcal F_x$ , since each such member contains a tail; conversely every tail belongs to $\mathcal F_x$ . 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 15 results over 6 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)
Filter (set theory) (Wikipedia) (standard reference, not scraped)
Net (mathematics) (Wikipedia) (standard reference, not scraped)