ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-02

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.

Every function from discrete $N$ to $[0, 1]$ extends uniquely to $β N$

Example

For the discrete space $N$ , every function $g : N \to [0, 1]$ extends uniquely to every Stone–Čech compactification $β N$ .

Facts & Assumptions

Given: A function $g : N \to [0, 1]$ and a Stone–Čech compactification $(β N, i)$ .

[L1]

Every map with discrete domain is continuous (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies).

[L2]

The interval $[0, 1]$ is compact and Hausdorff (Heine-Borel by bisection: every closed bounded interval $[a, b]$ is compact, Distinct points of a metric space have disjoint balls around them).

Verification

technique · direct

1.1

The map $g$ is continuous by [L1]. By [L2], the interval $[0, 1]$ is compact Hausdorff, so the extension and uniqueness clause of The Stone–Čech compactification by its compact-Hausdorff extension property gives a unique continuous $\overset{ˉ}{g} : β N \to [0, 1]$ with $\overset{ˉ}{g} \circ i = g$ .

L1L2

2.1

This is the asserted extension.

step 1.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

43 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

E. Moorhouse, The Stone–Čech Compactification (standard reference, not scraped)

Every function from discrete NN to [0,1][0,1] extends uniquely to βNβN