Alphabeta Math
ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-11
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.

For a fixed space XX, product with XX defines an endofunctor of Top\mathbf{Top}

Example

Fix a topological space XX. The assignment Y↦X×YY\mapsto X\times Y is an endofunctor of Top\mathbf{Top}.

Facts & Assumptions

Given: A fixed topological space XX.

[L1]

Topological spaces and continuous maps form Top\mathbf{Top} (Topological spaces and continuous maps form the large locally small category Top\mathbf{Top}).

[L2]

A map into a product is continuous exactly when its coordinate maps are continuous (A map into a product is continuous iff each of its components is; the projections are continuous and open; and each projection is surjective when every factor is nonempty, which for an infinite index set uses the Axiom of Choice).

[L3]

An endofunctor must preserve identities and composition (Covariant functor, identity functor, composite functor, and contravariant functor).

Verification

technique · direct
1.1

Define TX(Y)=X×YT_X(Y)=X\times Y with the product topology. For a continuous f:Y→Zf:Y\to Z, define TX(f)=1X×fT_X(f)=1_X\times f by (x,y)↦(x,f(y))(x,y)\mapsto(x,f(y)).

L1
2.1

Its coordinate maps are the first projection and ff after the second projection, so TX(f)T_X(f) is continuous by [L2].

step 1.1L2
2.2

Pointwise, 1X×1Y=1X×Y1_X\times1_Y=1_{X\times Y} and (1X×g)(1X×f)=1X×(gf)(1_X\times g)(1_X\times f)=1_X\times(gf).

step 1.1
3.1

Thus TXT_X sends every morphism of Top\mathbf{Top} to a morphism and obeys the two functor equations. It is an endofunctor by [L3].

step 2.1step 2.2L1L3∎

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: 38 results over 10 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