idealOfSet_ofIdeal_isClosed — Mathlib

English

Given f ∈ C(X,ℝ≥0) and c>0, there exists g ∈ C(X,ℝ≥0) such that (g f)(x) ≤ 1 for all x and the set {x | c ≤ f(x)} is contained in the equal-on-one set of (g f).

Русский

Для f ∈ C(X,ℝ≥0) и c>0 существует g ∈ C(X,ℝ≥0) так, что (g f)(x) ≤ 1 для всех x и множество {x | c ≤ f(x)} совпадает на области с 1 через g f.

LaTeX

$$$ \\exists g \\in C(X,\\mathbb{R}_{\\ge 0}), (\\forall x, (g f)(x) \\le 1) \\wedge \\{x : X \\mid c \\le f(x)\\} = {x : X \\mid (g f)(x) = 1} $$$

Lean4

theorem idealOfSet_ofIdeal_isClosed {I : Ideal C(X, 𝕜)} (hI : IsClosed (I : Set C(X, 𝕜))) :
    idealOfSet 𝕜 (setOfIdeal I) = I :=
  (idealOfSet_ofIdeal_eq_closure I).trans (Ideal.ext <| Set.ext_iff.mp hI.closure_eq)

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