File: Mathlib/Topology/ContinuousFunction/Ideals.lean

full_name: ContinuousMap.idealOfSet
code: /-- Given a topological ring `R` and `s : Set X`, construct the ideal in `C(X, R)` of functions
which vanish on the complement of `s`. -/
def idealOfSet (s : Set X) : Ideal C(X, R) 
goal: Ideal C(X, R)

full_name: ContinuousMap.idealOfSet_closed
code: theorem idealOfSet_closed [T2Space R] (s : Set X) :
    IsClosed (idealOfSet R s : Set C(X, R)) 
goal: IsClosed ↑(ContinuousMap.idealOfSet R s)

full_name: ContinuousMap.mem_idealOfSet
code: theorem mem_idealOfSet {s : Set X} {f : C(X, R)} :
    f ∈ idealOfSet R s ↔ ∀ ⦃x : X⦄, x ∈ sᶜ → f x = 0 
goal: f ∈ ContinuousMap.idealOfSet R s ↔ ∀ ⦃x : X⦄, x ∈ sᶜ → f x = 0

full_name: ContinuousMap.not_mem_idealOfSet
code: theorem not_mem_idealOfSet {s : Set X} {f : C(X, R)} : f ∉ idealOfSet R s ↔ ∃ x ∈ sᶜ, f x ≠ 0 
goal: f ∉ ContinuousMap.idealOfSet R s ↔ ∃ x ∈ sᶜ, f x ≠ 0

full_name: ContinuousMap.setOfIdeal
code: /-- Given an ideal `I` of `C(X, R)`, construct the set of points for which every function in the
ideal vanishes on the complement. -/
def setOfIdeal (I : Ideal C(X, R)) : Set X 
goal: Set X

full_name: ContinuousMap.not_mem_setOfIdeal
code: theorem not_mem_setOfIdeal {I : Ideal C(X, R)} {x : X} :
    x ∉ setOfIdeal I ↔ ∀ ⦃f : C(X, R)⦄, f ∈ I → f x = 0 
goal: x ∉ ContinuousMap.setOfIdeal I ↔ ∀ ⦃f : C(X, R)⦄, f ∈ I → f x = 0

full_name: ContinuousMap.mem_setOfIdeal
code: theorem mem_setOfIdeal {I : Ideal C(X, R)} {x : X} :
    x ∈ setOfIdeal I ↔ ∃ f ∈ I, (f : C(X, R)) x ≠ 0 
goal: x ∈ ContinuousMap.setOfIdeal I ↔ ∃ f ∈ I, f x ≠ 0

full_name: ContinuousMap.setOfIdeal_open
code: theorem setOfIdeal_open [T2Space R] (I : Ideal C(X, R)) : IsOpen (setOfIdeal I) 
goal: IsOpen (ContinuousMap.setOfIdeal I)

full_name: ContinuousMap.opensOfIdeal_coe
code: /-- The open set `ContinuousMap.setOfIdeal I` realized as a term of `opens X`. -/
@[simps]
def opensOfIdeal [T2Space R] (I : Ideal C(X, R)) : Opens X 
goal: ↑(ContinuousMap.opensOfIdeal I) = ContinuousMap.setOfIdeal I

full_name: ContinuousMap.opensOfIdeal
code: /-- The open set `ContinuousMap.setOfIdeal I` realized as a term of `opens X`. -/
@[simps]
def opensOfIdeal [T2Space R] (I : Ideal C(X, R)) : Opens X 
goal: TopologicalSpace.Opens X

full_name: ContinuousMap.setOfTop_eq_univ
code: @[simp]
theorem setOfTop_eq_univ [Nontrivial R] : setOfIdeal (⊤ : Ideal C(X, R)) = Set.univ 
goal: ContinuousMap.setOfIdeal ⊤ = Set.univ

full_name: ContinuousMap.idealOfEmpty_eq_bot
code: @[simp]
theorem idealOfEmpty_eq_bot : idealOfSet R (∅ : Set X) = ⊥ 
goal: ContinuousMap.idealOfSet R ∅ = ⊥

full_name: ContinuousMap.mem_idealOfSet_compl_singleton
code: @[simp]
theorem mem_idealOfSet_compl_singleton (x : X) (f : C(X, R)) :
    f ∈ idealOfSet R ({x}ᶜ : Set X) ↔ f x = 0 
goal: f ∈ ContinuousMap.idealOfSet R {x}ᶜ ↔ f x = 0

full_name: ContinuousMap.ideal_gc
code: theorem ideal_gc : GaloisConnection (setOfIdeal : Ideal C(X, R) → Set X) (idealOfSet R) 
goal: GaloisConnection ContinuousMap.setOfIdeal (ContinuousMap.idealOfSet R)

full_name: ContinuousMap.exists_mul_le_one_eqOn_ge
code: /-- An auxiliary lemma used in the proof of `ContinuousMap.idealOfSet_ofIdeal_eq_closure` which may
be useful on its own. -/
theorem exists_mul_le_one_eqOn_ge (f : C(X, ℝ≥0)) {c : ℝ≥0} (hc : 0 < c) :
    ∃ g : C(X, ℝ≥0), (∀ x : X, (g * f) x ≤ 1) ∧ {x : X | c ≤ f x}.EqOn (g * f) 1 
goal: ∃ g, (∀ (x : X), (g * f) x ≤ 1) ∧ Set.EqOn (⇑(g * f)) 1 {x | c ≤ f x}

full_name: ContinuousMap.idealOfSet_ofIdeal_eq_closure
code: @[simp]
theorem idealOfSet_ofIdeal_eq_closure (I : Ideal C(X, 𝕜)) :
    idealOfSet 𝕜 (setOfIdeal I) = I.closure 
goal: ContinuousMap.idealOfSet 𝕜 (ContinuousMap.setOfIdeal I) = I.closure

full_name: ContinuousMap.idealOfSet_ofIdeal_isClosed
code: theorem idealOfSet_ofIdeal_isClosed {I : Ideal C(X, 𝕜)} (hI : IsClosed (I : Set C(X, 𝕜))) :
    idealOfSet 𝕜 (setOfIdeal I) = I 
goal: ContinuousMap.idealOfSet 𝕜 (ContinuousMap.setOfIdeal I) = I

full_name: ContinuousMap.setOfIdeal_ofSet_eq_interior
code: @[simp]
theorem setOfIdeal_ofSet_eq_interior (s : Set X) : setOfIdeal (idealOfSet 𝕜 s) = interior s 
goal: ContinuousMap.setOfIdeal (ContinuousMap.idealOfSet 𝕜 s) = interior s

full_name: ContinuousMap.setOfIdeal_ofSet_of_isOpen
code: theorem setOfIdeal_ofSet_of_isOpen {s : Set X} (hs : IsOpen s) : setOfIdeal (idealOfSet 𝕜 s) = s 
goal: ContinuousMap.setOfIdeal (ContinuousMap.idealOfSet 𝕜 s) = s

full_name: ContinuousMap.idealOpensGI_choice
code: /-- The Galois insertion `ContinuousMap.opensOfIdeal : Ideal C(X, 𝕜) → Opens X` and
`fun s ↦ ContinuousMap.idealOfSet ↑s`. -/
@[simps]
def idealOpensGI :
    GaloisInsertion (opensOfIdeal : Ideal C(X, 𝕜) → Opens X) fun s => idealOfSet 𝕜 s 
goal: ∀ (x : ContinuousMap.idealOfSet 𝕜 ↑(ContinuousMap.opensOfIdeal I) ≤ I),
  (ContinuousMap.idealOpensGI X 𝕜).choice I x = ContinuousMap.opensOfIdeal I.closure

full_name: ContinuousMap.idealOpensGI
code: /-- The Galois insertion `ContinuousMap.opensOfIdeal : Ideal C(X, 𝕜) → Opens X` and
`fun s ↦ ContinuousMap.idealOfSet ↑s`. -/
@[simps]
def idealOpensGI :
    GaloisInsertion (opensOfIdeal : Ideal C(X, 𝕜) → Opens X) fun s => idealOfSet 𝕜 s 
goal: GaloisInsertion ContinuousMap.opensOfIdeal fun s => ContinuousMap.idealOfSet 𝕜 ↑s

full_name: ContinuousMap.idealOfSet_isMaximal_iff
code: theorem idealOfSet_isMaximal_iff (s : Opens X) :
    (idealOfSet 𝕜 (s : Set X)).IsMaximal ↔ IsCoatom s 
goal: (ContinuousMap.idealOfSet 𝕜 ↑s).IsMaximal ↔ IsCoatom s

full_name: ContinuousMap.idealOf_compl_singleton_isMaximal
code: theorem idealOf_compl_singleton_isMaximal (x : X) : (idealOfSet 𝕜 ({x}ᶜ : Set X)).IsMaximal 
goal: (ContinuousMap.idealOfSet 𝕜 {x}ᶜ).IsMaximal

full_name: ContinuousMap.setOfIdeal_eq_compl_singleton
code: theorem setOfIdeal_eq_compl_singleton (I : Ideal C(X, 𝕜)) [hI : I.IsMaximal] :
    ∃ x : X, setOfIdeal I = {x}ᶜ 
goal: ∃ x, ContinuousMap.setOfIdeal I = {x}ᶜ

full_name: ContinuousMap.ideal_isMaximal_iff
code: theorem ideal_isMaximal_iff (I : Ideal C(X, 𝕜)) [hI : IsClosed (I : Set C(X, 𝕜))] :
    I.IsMaximal ↔ ∃ x : X, idealOfSet 𝕜 {x}ᶜ = I 
goal: I.IsMaximal ↔ ∃ x, ContinuousMap.idealOfSet 𝕜 {x}ᶜ = I

full_name: WeakDual.CharacterSpace.continuousMapEval
code: /-- The natural continuous map from a locally compact topological space `X` to the
`WeakDual.characterSpace 𝕜 C(X, 𝕜)` which sends `x : X` to point evaluation at `x`. -/
def continuousMapEval : C(X, characterSpace 𝕜 C(X, 𝕜)) 
goal: C(X, ↑(WeakDual.characterSpace 𝕜 C(X, 𝕜)))

full_name: WeakDual.CharacterSpace.continuousMapEval_apply_apply
code: @[simp]
theorem continuousMapEval_apply_apply (x : X) (f : C(X, 𝕜)) : continuousMapEval X 𝕜 x f = f x 
goal: ((WeakDual.CharacterSpace.continuousMapEval X 𝕜) x) f = f x

full_name: WeakDual.CharacterSpace.continuousMapEval_bijective
code: theorem continuousMapEval_bijective : Bijective (continuousMapEval X 𝕜) 
goal: Function.Bijective ⇑(WeakDual.CharacterSpace.continuousMapEval X 𝕜)

full_name: WeakDual.CharacterSpace.homeoEval
code: /-- This is the natural homeomorphism between a compact Hausdorff space `X` and the
`WeakDual.characterSpace 𝕜 C(X, 𝕜)`. -/
noncomputable def homeoEval : X ≃ₜ characterSpace 𝕜 C(X, 𝕜) 
goal: X ≃ₜ ↑(WeakDual.characterSpace 𝕜 C(X, 𝕜))