continuous_rng_of_principal_iff_forall

English

There is a principal version of the topological equality linking restricted product topology with induced topology.

Русский

Принципиальное равенство для топологии ограниченного произведения через индуцированную топологію.

LaTeX

$$topologicalSpace_eq_of_principal : RestrictedProduct.topologicalSpace R A (Filter.principal S) = .induced (coe) inferInstance$$

Lean4

theorem continuous_rng_of_principal_iff_forall {X : Type*} [TopologicalSpace X] {f : X → Πʳ (i : ι), [R i, A i]_[𝓟 S]} :
    Continuous f ↔ ∀ i : ι, Continuous ((fun x ↦ x i) ∘ f) :=
  continuous_rng_of_principal.trans continuous_pi_iff

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