TheoremStatement: Literature-sourcedProof: 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 point is a cluster point of a net if and only if some subnet converges to it

Statement

For a net $x : D \to X$ and $p \in X$ , $p$ is a cluster point of $x$ if and only if $x$ has a subnet converging to $p$ .

Facts & Assumptions

Given: A net $x : D \to X$ in a topological space and a point $p \in X$ .

[A1]

A cluster point is one for which every neighbourhood is visited frequently, and convergence means eventual membership in every neighbourhood (Convergence and cluster points of a net in a topological space).

[A2]

Intersections of finitely many 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).

[A3]

A subnet is given by an eventually cofinal index map (Subnet via an eventually cofinal index map).

Proof

technique · constructive

1.1

Suppose $p$ is a cluster point. Let $E = {(d, N) : N \in N (p), d \in D, x_{d} \in N}$ , ordered by $(d, N) ⪯ (e, M)$ when $d \leq e$ and $M \subseteq N$ .

A1construct

1.2

Conversely, suppose a subnet $y_{e} = x_{ϕ (e)}$ converges to $p$ . Given a neighbourhood $N$ and $d \in D$ , choose $e_{0}$ after which $y$ lies in $N$ and choose $e_{1}$ after which $ϕ (e) \geq d$ ; a common upper bound $e$ of $e_{0}, e_{1}$ gives $ϕ (e) \geq d$ and $x_{ϕ (e)} = y_{e} \in N$ . Hence $x$ is frequently in $N$ .

A1A3

2.1

The set $E$ is directed: for $(d, N), (e, M) \in E$ , take $h \geq d, e$ in $D$ ; frequent membership in $N \cap M$ gives $k \geq h$ with $x_{k} \in N \cap M$ , and $(k, N \cap M)$ is above both pairs.

step 1.1A1A2

2.2

Put $y_{(d, N)} = x_{d}$ and $ϕ (d, N) = d$ . For every $d_{0} \in D$ , the pair $(d_{0}, X)$ lies in $E$ , and every later pair has first coordinate at least $d_{0}$ . Thus $ϕ$ is eventually cofinal and $y$ is a subnet of $x$ .

step 1.1A3

2.3

For a neighbourhood $N$ of $p$ , choose $(d, N) \in E$ using frequent membership in $N$ . Every pair later than it has second coordinate contained in $N$ , hence its $y$ -value lies in $N$ . Thus $y \to p$ .

step 1.1A1

3.1

Steps 1.1 and 2.1--2.3 construct a convergent subnet from a cluster point, and step 1.2 gives the converse.

step 2.2step 2.3step 1.2discharge-construct∎

Depends on

Used by

Dependency tree · two levels

7 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

Schlumprecht, Math 655 notes (standard reference, not scraped)
Net (mathematics) (Wikipedia) (standard reference, not scraped)