TheoremStatement: Literature-sourcedProof: 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 point lies in the closure of a set if and only if a net in the set converges to it

Statement

For $A\subseteq X$ and $p\in X$ , one has $p\in\overline A$ if and only if there is a net in $A$ converging to $p$ .

Facts & Assumptions

Given: A subset $A$ of a topological space $X$ and a point $p\in X$ .

[L1]

$p\in\overline A$ exactly when every neighbourhood of $p$ meets $A$ (A point lies in the closure of $A$ iff every basic neighbourhood of it meets $A$ ; the closure is the smallest closed superset and equals $A$ together with its derived set).

[L2]

Finite intersections of neighbourhoods of $p$ are neighbourhoods of $p$ (Neighbourhood of a point and neighbourhood base, with this library's convention that a neighbourhood need not be open).

[L3]

A net converges exactly when it is eventually in every neighbourhood (Convergence and cluster points of a net in a topological space).

Proof

technique · constructive

1.1

Suppose $p\in\overline A$ . Let $E=\{(N,a):N\in\mathcal N(p),\ a\in N\cap A\}$ , ordered by $(N,a)\preceq(M,b)$ when $M\subseteq N$ , and put $x_{(N,a)}=a$ .

L1construct

1.2

Conversely, if a net $x$ in $A$ converges to $p$ , every neighbourhood $N$ of $p$ contains some eventual value $x_d\in A$ , so $N\cap A\ne\varnothing$ and $p\in\overline A$ .

L1L3

2.1

The index set is directed: for $(N,a),(M,b)$ , the set $N\cap M$ is a neighbourhood and meets $A$ ; for $c\in(N\cap M)\cap A$ , the pair $(N\cap M,c)$ is above both.

step 1.1L1L2

2.2

Given a neighbourhood $N$ of $p$ , choose $(N,a)\in E$ . Every later pair has its second coordinate in a subset of $N$ , so $x$ is eventually in $N$ and therefore converges to $p$ .

step 1.1L3

3.1

Steps 1.1--2.1 construct the required net and step 1.2 proves the converse.

step 2.2step 1.2discharge-construct∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 9 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)
Net (mathematics) (Wikipedia) (standard reference, not scraped)
net (nLab) (standard reference, not scraped)