Alphabeta Math
DefinitionDefinition: AI-adaptedProof: Not applicableaudited 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.

A family of continuous unit-interval-valued functions that separates points from closed sets

Definition

Let X be a topological space. A family F of continuous maps f:X→[0,1] (Continuity of a map of topological spaces at a point and globally, Intervals of R: the nine order-convex forms, nondegeneracy, and length) separates points from closed sets when both conditions hold:

  1. for every distinct x,y∈X, some f∈F has f(x)≠f(y); and
  2. for every closed C⊆X and every x∉C, some f∈F satisfies f(x)=1 and f[C]={0}.

The empty family has these properties precisely when X=∅. Indeed, if x∈X, the closed set C=∅ makes clause 2 demand a member of the family. Thus a one-point space still needs a separating function for its point and the empty closed set; no coordinate is silently selected in either case.

Depends on

Used by

Dependency tree · two levels

9 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