ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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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.

In the cocountable topology on $\mathbb{R}$ , a closure point outside $[0,1]$ is reached by a net in $[0,1]$ but by no sequence in $[0,1]$

Example

Give $\mathbb R$ the cocountable topology, let $A=[0,1]$ , and let $p=2$ . Then $p\in\overline A$ , hence a net in $A$ converges to $p$ , but no sequence in $A$ converges to $p$ .

Facts & Assumptions

Given: The cocountable topology on $\mathbb R$ , $A=[0,1]$ , and $p=2$ .

[L1]

Nonempty cocountable opens have at most countable complements (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies, Finite, countably infinite, countable, uncountable).

[L2]

$[0,1]$ is uncountable (Every nondegenerate interval of $\mathbb{R}$ is uncountable).

[L3]

A sequence converges only if it is eventually in every neighbourhood of its proposed limit (Convergence and cluster points of a sequence in a topological space, sequential continuity, and the sequential closure).

[L4]

A point lies in the closure of a subset exactly when some net in that subset converges to it (A point lies in the closure of a set if and only if a net in the set converges to it).

Verification

technique · constructive

1.1

Every neighbourhood $N$ of $2$ has at most countable complement, so it meets the uncountable set $A$ . Hence $2\in\overline A$ , and [L4] supplies a net in $A$ converging to $2$ .

L1L2L4construct

1.2

Let $(a_n)$ be a sequence in $A$ . Its range is at most countable and omits $2$ , so $\mathbb R\setminus\{a_n:n\in\mathbb N\}$ is a neighbourhood of $2$ containing none of its terms. Thus $(a_n)$ does not converge to $2$ .

L1L3

2.1

The net from step 1.1 detects the closure point, whereas no sequence in $A$ does.

step 1.1step 1.2discharge-construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 86 results over 18 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

Cocountable topology (Wikipedia) (standard reference, not scraped)
Net (mathematics) (Wikipedia) (standard reference, not scraped)

In the cocountable topology on R\mathbb{R}R, a closure point outside [0,1][0,1][0,1] is reached by a net in [0,1][0,1][0,1] but by no sequence in [0,1][0,1][0,1]