DefinitionDefinition: Literature-sourcedProof: Not applicablejudge 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.

Directed preorders and nets

Definition

A directed preorder is a nonempty set $D$ with a reflexive, transitive relation $\leq$ such that every $d, e \in D$ have a common upper bound: some $f \in D$ satisfies $d \leq f$ and $e \leq f$ . Antisymmetry is not required; thus this is a preorder obtained by omitting antisymmetry from the partial-order axioms of Partial order and partially ordered set.

If $X$ is the underlying set of a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison), a net in $X$ indexed by $D$ is a function $x : D \to X$ , written $(x_{d})_{d \in D}$ . The order on $D$ records which indices are sufficiently far along; it need not be a linear order.

Remarks

Some texts require a directed set to be a partial order. The present preorder convention is deliberate: none of the convergence arguments needs antisymmetry, and it permits convenient index systems with equivalent stages.

Depends on

Used by

A directed set and an inverse system of groups indexed by it Definition
A net is eventually or frequently in a subset of its codomain Definition
Convergence and cluster points of a net in a topological space Definition
Square-summable families on an arbitrary index set and the space ℓ²(I) Definition
Subnet via an eventually cofinal index map Definition
The canonical net indexed by the pairs (A,x) with A in a filter and x∈ A Definition
The tail filter of a net Definition
Weak convergence of nets and sequences Definition
Weak star convergence Definition
A neighbourhood-indexed net in A converges to each point of Ā Example
Finite partial sums of a real family form a net directed by inclusion Example
Fourier series converge in mean square Theorem

Dependency tree · one level

2 results within one dependency step 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)
Directed set (Wikipedia) (standard reference, not scraped)
Net (mathematics) (Wikipedia) (standard reference, not scraped)
net (nLab) (standard reference, not scraped)