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ExampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02
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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.
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Every function from discrete N\mathbb N to [0,1][0,1] extends uniquely to βN\beta\mathbb N

Example

For the discrete space N\mathbb N, every function g:N→[0,1]g:\mathbb N\to[0,1] extends uniquely to every Stone–Čech compactification βN\beta\mathbb N.

Facts & Assumptions

Given: A function g:N→[0,1]g:\mathbb N\to[0,1] and a Stone–Čech compactification (βN,i)(\beta\mathbb 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][0,1] is compact and Hausdorff (Heine-Borel by bisection: every closed bounded interval [a,b][a,b] is compact, Distinct points of a metric space have disjoint balls around them).

Verification

technique · direct
1.1

The map gg is continuous by [L1]. By [L2], the interval [0,1][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 gˉ:βN→[0,1]\bar g:\beta\mathbb N\to[0,1] with gˉ∘i=g\bar 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 91 results over 24 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