Alphabeta Math
DefinitionDefinition: AI-adaptedProof: Not applicableverified 2026-08-10 (gpt-5.6-terra-codex-subscription)
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.

The space C(K,R) of continuous real-valued functions on a nonempty compact metric space

Definition

Let (K,d) be a nonempty compact metric space (Open cover, subcover, compact metric space, and compact subset of a metric space). Define

C(K,R):={ f∈RK:f:(K,d)→(R,dR) is continuous },

where RK is the function space of The vector space FX of all functions X→F with pointwise operations, and Fn as the case X=n={0,1,…,n−1} and dR(s,t)=∣s−t∣ is the usual metric (The absolute value makes R a metric space: d(x,y)=∣x−y∣ is a metric, its open balls are the intervals (x−r,x+r), and it is unbounded, Continuity of a map between metric spaces, at a point and globally, in the ε-δ form).

This definition introduces the set of continuous functions only. Boundedness and the supremum metric are assertions to be proved, not clauses of the definition.

Depends on

Dependency tree · 2 levels

Sources