@[simp]
theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
    idealOfSet 𝕜 (setOfIdeal I) = I.closure := by
  /- Since `idealOfSet 𝕜 (setOfIdeal I)` is closed and contains `I`, it contains `I.closure`.
    For the reverse inclusion, given `f ∈ idealOfSet 𝕜 (setOfIdeal I)` and `(ε : ℝ≥0) > 0` it
    suffices to show that `f` is within `ε` of `I`.-/
  refine' le_antisymm _
      ((idealOfSet_closed 𝕜 <| setOfIdeal I).closure_subset_iff.mpr fun f hf x hx =>
        not_mem_setOfIdeal.mp hx hf)
  refine' (fun f hf => Metric.mem_closure_iff.mpr fun ε hε => _)
  lift ε to ℝ≥0 using hε.lt.le
  replace hε := show (0 : ℝ≥0) < ε from hε
  simp_rw [dist_nndist]
  norm_cast
  -- Let `t := {x : X | ε / 2 ≤ ‖f x‖₊}}` which is closed and disjoint from `set_of_ideal I`.
  set t := {x : X | ε / 2 ≤ ‖f x‖₊}
  have ht : IsClosed t := isClosed_le continuous_const (map_continuous f).nnnorm
  have htI : Disjoint t (setOfIdeal I)ᶜ := by
    refine' Set.subset_compl_iff_disjoint_left.mp fun x hx => _
    simpa only [Set.mem_setOf, Set.mem_compl_iff, not_le] using
      (nnnorm_eq_zero.mpr (mem_idealOfSet.mp hf hx)).trans_lt (half_pos hε)
  /- It suffices to produce `g : C(X, ℝ≥0)` which takes values in `[0,1]` and is constantly `1` on
    `t` such that when composed with the natural embedding of `ℝ≥0` into `𝕜` lies in the ideal `I`.
    Indeed, then `‖f - f * ↑g‖ ≤ ‖f * (1 - ↑g)‖ ≤ ⨆ ‖f * (1 - ↑g) x‖`. When `x ∉ t`, `‖f x‖ < ε / 2`
    and `‖(1 - ↑g) x‖ ≤ 1`, and when `x ∈ t`, `(1 - ↑g) x = 0`, and clearly `f * ↑g ∈ I`. -/
  suffices
    ∃ g : C(X, ℝ≥0), (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g ∈ I ∧ (∀ x, g x ≤ 1) ∧ t.EqOn g 1 by
    obtain ⟨g, hgI, hg, hgt⟩ := this
    refine' ⟨f * (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g, I.mul_mem_left f hgI, _⟩
    rw [nndist_eq_nnnorm]
    refine' (nnnorm_lt_iff _ hε).2 fun x => _
    simp only [coe_sub, coe_mul, Pi.sub_apply, Pi.mul_apply]
    by_cases hx : x ∈ t
    · simpa only [hgt hx, comp_apply, Pi.one_apply, ContinuousMap.coe_coe, algebraMapClm_apply,
        map_one, mul_one, sub_self, nnnorm_zero] using hε
    · refine' lt_of_le_of_lt _ (half_lt_self hε)
      have :=
        calc
          ‖((1 - (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x : 𝕜)‖₊ =
              ‖1 - algebraMap ℝ≥0 𝕜 (g x)‖₊ := by
            simp only [coe_sub, coe_one, coe_comp, ContinuousMap.coe_coe, Pi.sub_apply,
              Pi.one_apply, Function.comp_apply, algebraMapClm_apply]
          _ = ‖algebraMap ℝ≥0 𝕜 (1 - g x)‖₊ := by
            simp only [Algebra.algebraMap_eq_smul_one, NNReal.smul_def, ge_iff_le,
              NNReal.coe_sub (hg x), NNReal.coe_one, sub_smul, one_smul]
          _ ≤ 1 := (nnnorm_algebraMap_nNReal 𝕜 (1 - g x)).trans_le tsub_le_self
      calc
        ‖f x - f x * (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g x‖₊ =
            ‖f x * (1 - (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ := by
          simp only [mul_sub, coe_sub, coe_one, Pi.sub_apply, Pi.one_apply, mul_one]
        _ ≤ ε / 2 * ‖(1 - (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g) x‖₊ :=
          ((nnnorm_mul_le _ _).trans
            (mul_le_mul_right' (not_le.mp <| show ¬ε / 2 ≤ ‖f x‖₊ from hx).le _))
        _ ≤ ε / 2 := by simpa only [mul_one] using mul_le_mul_left' this _
  /- There is some `g' : C(X, ℝ≥0)` which is strictly positive on `t` such that the composition
    `↑g` with the natural embedding of `ℝ≥0` into `𝕜` lies in `I`. This follows from compactness of
    `t` and that we can do it in any neighborhood of a point `x ∈ t`. Indeed, since `x ∈ t`, then
    `fₓ x ≠ 0` for some `fₓ ∈ I` and so `fun y ↦ ‖(star fₓ * fₓ) y‖₊` is strictly posiive in a
    neighborhood of `y`. Moreover, `(‖(star fₓ * fₓ) y‖₊ : 𝕜) = (star fₓ * fₓ) y`, so composition of
    this map with the natural embedding is just `star fₓ * fₓ ∈ I`. -/
  have : ∃ g' : C(X, ℝ≥0), (algebraMapClm ℝ≥0 𝕜 : C(ℝ≥0, 𝕜)).comp g' ∈ I ∧ ∀ x ∈ t, 0 < g' x := by
    refine' ht.isCompact.induction_on _ _ _ _
    · refine' ⟨0, _, fun x hx => False.elim hx⟩
      convert I.zero_mem
      ext
      simp only [comp_apply, zero_apply, ContinuousMap.coe_coe, map_zero]
    · rintro s₁ s₂ hs ⟨g, hI, hgt⟩; exact ⟨g, hI, fun x hx => hgt x (hs hx)⟩
    · 

Proof state:
case refine'_3
X : Type u_1
𝕜 : Type u_2
inst✝³ : IsROrC 𝕜
inst✝² : TopologicalSpace X
inst✝¹ : CompactSpace X
inst✝ : T2Space X
I : Ideal C(X, 𝕜)
f : C(X, 𝕜)
hf : f ∈ idealOfSet 𝕜 (setOfIdeal I)
ε : ℝ≥0
hε : 0 < ε
t : Set X := {x | ε / 2 ≤ ‖f x‖₊}
ht : IsClosed t
htI : Disjoint t (setOfIdeal I)ᶜ
⊢ ∀ ⦃s t : Set X⦄,
    (∃ g', comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I ∧ ∀ x ∈ s, 0 < g' x) →
      (∃ g', comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I ∧ ∀ x ∈ t, 0 < g' x) →
        ∃ g', comp (↑(algebraMapClm ℝ≥0 𝕜)) g' ∈ I ∧ ∀ x ∈ s ∪ t, 0 < g' x

Next tactic:
rintro s₁ s₂ ⟨g₁, hI₁, hgt₁⟩ ⟨g₂, hI₂, hgt₂⟩