countable_hom — Mathlib

English

If M is FG over L and N is countable, then the set of L-homomorphisms M→[L]N is countable.

Русский

Пусть M порождается конечным образом; тогда множество гомоморфизмов L от M к счетному N конечно счетно.

LaTeX

$$$\text{FG}(L,M) \land \text{Countable}(N) \Rightarrow |M \to_{L} N| < \infty$$$

Lean4

theorem countable_hom (N : Type*) [L.Structure N] [Countable N] (h : FG L M) : Countable (M →[L] N) :=
  by
  let ⟨S, finite_S, closure_S⟩ := fg_iff.1 h
  let g : (M →[L] N) → (S → N) := fun f ↦ f ∘ (↑)
  have g_inj : Function.Injective g := by
    intro f f' h
    apply Hom.eq_of_eqOn_dense closure_S
    intro x x_in_S
    exact congr_fun h ⟨x, x_in_S⟩
  have : Finite ↑S := (S.finite_coe_iff).2 finite_S
  exact Function.Embedding.countable ⟨g, g_inj⟩

Похожие утверждения