English
The topology on Hom(A, R) is the coarsest topology that makes the evaluation maps at every a ∈ A continuous; equivalently, it is the subspace topology from a polynomial presentation of A.
Русский
Топология на Hom(A, R) является наименьшей топологией, делающей отображения ev_a(f)=f(a) непрерывными для всех a ∈ A; эквивалентно субпространственной топологии от представления A как полинома.
LaTeX
$$$\mathcal{T}_{\mathrm{Hom}(A,R)} = \operatorname{induced}\left(\{\mathrm{ev}_a : \mathrm{Hom}(A,R) \to R \mid a \in A\}\right)$$$
Lean4
/-- The topology on `Hom(A, R)` for a topological ring `R`, given by the coarsest topology that
makes `f ↦ f x` continuous for all `x : A` (see `continuous_apply`).
Alternatively, given a presentation `A = ℤ[xᵢ]/I`,
this is the subspace topology `Hom(A, R) ↪ Hom(ℤ[xᵢ], R) = Rᶥ` (see `mvPolynomialHomeomorph`).
This is a scoped instance in `CommRingCat.HomTopology`.
-/
scoped instance : TopologicalSpace (A ⟶ R) :=
.induced (fun f ↦ f.hom : _ → A → R) inferInstance
