[GOAL]
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : Finite ι
inst✝ : (i : ι) → TopologicalSpace (π i)
h : PseudoMetrizableSpace X
⊢ FirstCountableTopology X
[PROOFSTEP]
rcases h with ⟨_, hm⟩
[GOAL]
case mk.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : Finite ι
inst✝ : (i : ι) → TopologicalSpace (π i)
w✝ : PseudoMetricSpace X
hm : UniformSpace.toTopologicalSpace = inst✝³
⊢ FirstCountableTopology X
[PROOFSTEP]
rw [← hm]
[GOAL]
case mk.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝³ : TopologicalSpace X
inst✝² : TopologicalSpace Y
inst✝¹ : Finite ι
inst✝ : (i : ι) → TopologicalSpace (π i)
w✝ : PseudoMetricSpace X
hm : UniformSpace.toTopologicalSpace = inst✝³
⊢ FirstCountableTopology X
[PROOFSTEP]
exact @UniformSpace.firstCountableTopology X PseudoMetricSpace.toUniformSpace EMetric.instIsCountablyGeneratedUniformity
[GOAL]
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : Finite ι
inst✝¹ : (i : ι) → TopologicalSpace (π i)
inst✝ : ∀ (i : ι), PseudoMetrizableSpace (π i)
⊢ PseudoMetrizableSpace ((i : ι) → π i)
[PROOFSTEP]
cases nonempty_fintype ι
[GOAL]
case intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : Finite ι
inst✝¹ : (i : ι) → TopologicalSpace (π i)
inst✝ : ∀ (i : ι), PseudoMetrizableSpace (π i)
val✝ : Fintype ι
⊢ PseudoMetrizableSpace ((i : ι) → π i)
[PROOFSTEP]
letI := fun i => pseudoMetrizableSpacePseudoMetric (π i)
[GOAL]
case intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : Finite ι
inst✝¹ : (i : ι) → TopologicalSpace (π i)
inst✝ : ∀ (i : ι), PseudoMetrizableSpace (π i)
val✝ : Fintype ι
this : (i : ι) → PseudoMetricSpace (π i) := fun i => pseudoMetrizableSpacePseudoMetric (π i)
⊢ PseudoMetrizableSpace ((i : ι) → π i)
[PROOFSTEP]
infer_instance
[GOAL]
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : Finite ι
inst✝¹ : (i : ι) → TopologicalSpace (π i)
inst✝ : ∀ (i : ι), MetrizableSpace (π i)
⊢ MetrizableSpace ((i : ι) → π i)
[PROOFSTEP]
cases nonempty_fintype ι
[GOAL]
case intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : Finite ι
inst✝¹ : (i : ι) → TopologicalSpace (π i)
inst✝ : ∀ (i : ι), MetrizableSpace (π i)
val✝ : Fintype ι
⊢ MetrizableSpace ((i : ι) → π i)
[PROOFSTEP]
letI := fun i => metrizableSpaceMetric (π i)
[GOAL]
case intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : Finite ι
inst✝¹ : (i : ι) → TopologicalSpace (π i)
inst✝ : ∀ (i : ι), MetrizableSpace (π i)
val✝ : Fintype ι
this : (i : ι) → MetricSpace (π i) := fun i => metrizableSpaceMetric (π i)
⊢ MetrizableSpace ((i : ι) → π i)
[PROOFSTEP]
infer_instance
[GOAL]
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : Finite ι
inst✝¹ : (i : ι) → TopologicalSpace (π i)
inst✝ : PseudoMetrizableSpace X
s : Set X
hs : IsSeparable s
⊢ SecondCountableTopology ↑s
[PROOFSTEP]
letI := pseudoMetrizableSpacePseudoMetric X
[GOAL]
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : Finite ι
inst✝¹ : (i : ι) → TopologicalSpace (π i)
inst✝ : PseudoMetrizableSpace X
s : Set X
hs : IsSeparable s
this : PseudoMetricSpace X := pseudoMetrizableSpacePseudoMetric X
⊢ SecondCountableTopology ↑s
[PROOFSTEP]
have := hs.separableSpace
[GOAL]
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁴ : TopologicalSpace X
inst✝³ : TopologicalSpace Y
inst✝² : Finite ι
inst✝¹ : (i : ι) → TopologicalSpace (π i)
inst✝ : PseudoMetrizableSpace X
s : Set X
hs : IsSeparable s
this✝ : PseudoMetricSpace X := pseudoMetrizableSpacePseudoMetric X
this : SeparableSpace ↑s
⊢ SecondCountableTopology ↑s
[PROOFSTEP]
exact UniformSpace.secondCountable_of_separable s
[GOAL]
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
⊢ ∃ f, Embedding f
[PROOFSTEP]
haveI : NormalSpace X := normalSpaceOfT3SecondCountable X
[GOAL]
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this : NormalSpace X
⊢ ∃ f, Embedding f
[PROOFSTEP]
rcases exists_countable_basis X with ⟨B, hBc, -, hB⟩
[GOAL]
case intro.intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
⊢ ∃ f, Embedding f
[PROOFSTEP]
let s : Set (Set X × Set X) :=
  {UV ∈ B ×ˢ B | closure UV.1 ⊆ UV.2}
    -- `s` is a countable set.
[GOAL]
case intro.intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
⊢ ∃ f, Embedding f
[PROOFSTEP]
haveI : Encodable s := ((hBc.prod hBc).mono (inter_subset_left _ _)).toEncodable
[GOAL]
case intro.intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝ : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this : Encodable ↑s
⊢ ∃ f, Embedding f
[PROOFSTEP]
letI : TopologicalSpace s := ⊥
[GOAL]
case intro.intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝¹ : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝ : Encodable ↑s
this : TopologicalSpace ↑s := ⊥
⊢ ∃ f, Embedding f
[PROOFSTEP]
haveI : DiscreteTopology s := ⟨rfl⟩
[GOAL]
case intro.intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
⊢ ∃ f, Embedding f
[PROOFSTEP]
rsuffices ⟨f, hf⟩ : ∃ f : X → s →ᵇ ℝ, Embedding f
[GOAL]
case intro.intro.intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
f : X → ↑s →ᵇ ℝ
hf : Embedding f
⊢ ∃ f, Embedding f
[PROOFSTEP]
exact
  ⟨fun x => (f x).extend (Encodable.encode' s) 0,
    (BoundedContinuousFunction.isometry_extend (Encodable.encode' s) (0 : ℕ →ᵇ ℝ)).embedding.comp hf⟩
[GOAL]
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
⊢ ∃ f, Embedding f
[PROOFSTEP]
have hd : ∀ UV : s, Disjoint (closure UV.1.1) UV.1.2ᶜ := fun UV =>
  disjoint_compl_right.mono_right
    (compl_subset_compl.2 UV.2.2)
      -- Choose a sequence of `εₙ > 0`, `n : s`, that is bounded above by `1` and tends to zero
        -- along the `cofinite` filter.
[GOAL]
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
⊢ ∃ f, Embedding f
[PROOFSTEP]
obtain ⟨ε, ε01, hε⟩ : ∃ ε : s → ℝ, (∀ UV, ε UV ∈ Ioc (0 : ℝ) 1) ∧ Tendsto ε cofinite (𝓝 0) :=
  by
  rcases posSumOfEncodable zero_lt_one s with ⟨ε, ε0, c, hεc, hc1⟩
  refine' ⟨ε, fun UV => ⟨ε0 UV, _⟩, hεc.summable.tendsto_cofinite_zero⟩
  exact (le_hasSum hεc UV fun _ _ => (ε0 _).le).trans hc1
[GOAL]
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
⊢ ∃ ε, (∀ (UV : ↑s), ε UV ∈ Ioc 0 1) ∧ Tendsto ε cofinite (𝓝 0)
[PROOFSTEP]
rcases posSumOfEncodable zero_lt_one s with ⟨ε, ε0, c, hεc, hc1⟩
[GOAL]
case mk.intro.intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε0 : ∀ (i : ↑s), 0 < ε i
c : ℝ
hεc : HasSum ε c
hc1 : c ≤ 1
⊢ ∃ ε, (∀ (UV : ↑s), ε UV ∈ Ioc 0 1) ∧ Tendsto ε cofinite (𝓝 0)
[PROOFSTEP]
refine' ⟨ε, fun UV => ⟨ε0 UV, _⟩, hεc.summable.tendsto_cofinite_zero⟩
[GOAL]
case mk.intro.intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε0 : ∀ (i : ↑s), 0 < ε i
c : ℝ
hεc : HasSum ε c
hc1 : c ≤ 1
UV : ↑s
⊢ ε UV ≤ 1
[PROOFSTEP]
exact (le_hasSum hεc UV fun _ _ => (ε0 _).le).trans hc1
[GOAL]
case intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
⊢ ∃ f, Embedding f
[PROOFSTEP]
have : ∀ UV : s, ∃ f : C(X, ℝ), EqOn f 0 UV.1.1 ∧ EqOn f (fun _ => ε UV) UV.1.2ᶜ ∧ ∀ x, f x ∈ Icc 0 (ε UV) :=
  by
  intro UV
  rcases exists_continuous_zero_one_of_closed isClosed_closure (hB.isOpen UV.2.1.2).isClosed_compl (hd UV) with
    ⟨f, hf₀, hf₁, hf01⟩
  exact
    ⟨ε UV • f, fun x hx => by simp [hf₀ (subset_closure hx)], fun x hx => by simp [hf₁ hx], fun x =>
      ⟨mul_nonneg (ε01 _).1.le (hf01 _).1, mul_le_of_le_one_right (ε01 _).1.le (hf01 _).2⟩⟩
[GOAL]
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
⊢ ∀ (UV : ↑s), ∃ f, EqOn (↑f) 0 (↑UV).fst ∧ EqOn (↑f) (fun x => ε UV) (↑UV).sndᶜ ∧ ∀ (x : X), ↑f x ∈ Icc 0 (ε UV)
[PROOFSTEP]
intro UV
[GOAL]
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
UV : ↑s
⊢ ∃ f, EqOn (↑f) 0 (↑UV).fst ∧ EqOn (↑f) (fun x => ε UV) (↑UV).sndᶜ ∧ ∀ (x : X), ↑f x ∈ Icc 0 (ε UV)
[PROOFSTEP]
rcases exists_continuous_zero_one_of_closed isClosed_closure (hB.isOpen UV.2.1.2).isClosed_compl (hd UV) with
  ⟨f, hf₀, hf₁, hf01⟩
[GOAL]
case intro.intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
UV : ↑s
f : C(X, ℝ)
hf₀ : EqOn (↑f) 0 (closure (↑UV).fst)
hf₁ : EqOn (↑f) 1 (↑UV).sndᶜ
hf01 : ∀ (x : X), ↑f x ∈ Icc 0 1
⊢ ∃ f, EqOn (↑f) 0 (↑UV).fst ∧ EqOn (↑f) (fun x => ε UV) (↑UV).sndᶜ ∧ ∀ (x : X), ↑f x ∈ Icc 0 (ε UV)
[PROOFSTEP]
exact
  ⟨ε UV • f, fun x hx => by simp [hf₀ (subset_closure hx)], fun x hx => by simp [hf₁ hx], fun x =>
    ⟨mul_nonneg (ε01 _).1.le (hf01 _).1, mul_le_of_le_one_right (ε01 _).1.le (hf01 _).2⟩⟩
[GOAL]
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
UV : ↑s
f : C(X, ℝ)
hf₀ : EqOn (↑f) 0 (closure (↑UV).fst)
hf₁ : EqOn (↑f) 1 (↑UV).sndᶜ
hf01 : ∀ (x : X), ↑f x ∈ Icc 0 1
x : X
hx : x ∈ (↑UV).fst
⊢ ↑(ε UV • f) x = OfNat.ofNat 0 x
[PROOFSTEP]
simp [hf₀ (subset_closure hx)]
[GOAL]
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
UV : ↑s
f : C(X, ℝ)
hf₀ : EqOn (↑f) 0 (closure (↑UV).fst)
hf₁ : EqOn (↑f) 1 (↑UV).sndᶜ
hf01 : ∀ (x : X), ↑f x ∈ Icc 0 1
x : X
hx : x ∈ (↑UV).sndᶜ
⊢ ↑(ε UV • f) x = (fun x => ε UV) x
[PROOFSTEP]
simp [hf₁ hx]
[GOAL]
case intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝³ : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝² : Encodable ↑s
this✝¹ : TopologicalSpace ↑s := ⊥
this✝ : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
this : ∀ (UV : ↑s), ∃ f, EqOn (↑f) 0 (↑UV).fst ∧ EqOn (↑f) (fun x => ε UV) (↑UV).sndᶜ ∧ ∀ (x : X), ↑f x ∈ Icc 0 (ε UV)
⊢ ∃ f, Embedding f
[PROOFSTEP]
choose f hf0 hfε hf0ε using this
[GOAL]
case intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
⊢ ∃ f, Embedding f
[PROOFSTEP]
have hf01 : ∀ UV x, f UV x ∈ Icc (0 : ℝ) 1 := fun UV x =>
  Icc_subset_Icc_right (ε01 _).2
    (hf0ε _ _)
      -- The embedding is given by `F x UV = f UV x`.
[GOAL]
case intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
⊢ ∃ f, Embedding f
[PROOFSTEP]
set F : X → s →ᵇ ℝ := fun x =>
  ⟨⟨fun UV => f UV x, continuous_of_discreteTopology⟩, 1, fun UV₁ UV₂ =>
    Real.dist_le_of_mem_Icc_01 (hf01 _ _) (hf01 _ _)⟩
[GOAL]
case intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
⊢ ∃ f, Embedding f
[PROOFSTEP]
have hF : ∀ x UV, F x UV = f UV x := fun _ _ => rfl
[GOAL]
case intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
⊢ ∃ f, Embedding f
[PROOFSTEP]
refine' ⟨F, Embedding.mk' _ (fun x y hxy => _) fun x => le_antisymm _ _⟩
[GOAL]
case intro.intro.refine'_1
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x y : X
hxy : F x = F y
⊢ x = y
[PROOFSTEP]
by_contra Hne
[GOAL]
case intro.intro.refine'_1
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x y : X
hxy : F x = F y
Hne : ¬x = y
⊢ False
[PROOFSTEP]
rcases hB.mem_nhds_iff.1 (isOpen_ne.mem_nhds Hne) with ⟨V, hVB, hxV, hVy⟩
[GOAL]
case intro.intro.refine'_1.intro.intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x y : X
hxy : F x = F y
Hne : ¬x = y
V : Set X
hVB : V ∈ B
hxV : x ∈ V
hVy : V ⊆ {y_1 | y_1 ≠ y}
⊢ False
[PROOFSTEP]
rcases hB.exists_closure_subset (hB.mem_nhds hVB hxV) with ⟨U, hUB, hxU, hUV⟩
[GOAL]
case intro.intro.refine'_1.intro.intro.intro.intro.intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x y : X
hxy : F x = F y
Hne : ¬x = y
V : Set X
hVB : V ∈ B
hxV : x ∈ V
hVy : V ⊆ {y_1 | y_1 ≠ y}
U : Set X
hUB : U ∈ B
hxU : x ∈ U
hUV : closure U ⊆ V
⊢ False
[PROOFSTEP]
set UV : ↥s := ⟨(U, V), ⟨hUB, hVB⟩, hUV⟩
[GOAL]
case intro.intro.refine'_1.intro.intro.intro.intro.intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x y : X
hxy : F x = F y
Hne : ¬x = y
V : Set X
hVB : V ∈ B
hxV : x ∈ V
hVy : V ⊆ {y_1 | y_1 ≠ y}
U : Set X
hUB : U ∈ B
hxU : x ∈ U
hUV : closure U ⊆ V
UV : ↑s := { val := (U, V), property := (_ : (U, V) ∈ B ×ˢ B ∧ closure (U, V).fst ⊆ (U, V).snd) }
⊢ False
[PROOFSTEP]
apply (ε01 UV).1.ne
[GOAL]
case intro.intro.refine'_1.intro.intro.intro.intro.intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x y : X
hxy : F x = F y
Hne : ¬x = y
V : Set X
hVB : V ∈ B
hxV : x ∈ V
hVy : V ⊆ {y_1 | y_1 ≠ y}
U : Set X
hUB : U ∈ B
hxU : x ∈ U
hUV : closure U ⊆ V
UV : ↑s := { val := (U, V), property := (_ : (U, V) ∈ B ×ˢ B ∧ closure (U, V).fst ⊆ (U, V).snd) }
⊢ 0 = ε UV
[PROOFSTEP]
calc
  (0 : ℝ) = F x UV := (hf0 UV hxU).symm
  _ = F y UV := by rw [hxy]
  _ = ε UV := hfε UV fun h : y ∈ V => hVy h rfl
[GOAL]
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x y : X
hxy : F x = F y
Hne : ¬x = y
V : Set X
hVB : V ∈ B
hxV : x ∈ V
hVy : V ⊆ {y_1 | y_1 ≠ y}
U : Set X
hUB : U ∈ B
hxU : x ∈ U
hUV : closure U ⊆ V
UV : ↑s := { val := (U, V), property := (_ : (U, V) ∈ B ×ˢ B ∧ closure (U, V).fst ⊆ (U, V).snd) }
⊢ ↑(F x) UV = ↑(F y) UV
[PROOFSTEP]
rw [hxy]
[GOAL]
case intro.intro.refine'_2
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
⊢ comap F (𝓝 (F x)) ≤ 𝓝 x
[PROOFSTEP]
refine' ((nhds_basis_ball.comap _).le_basis_iff hB.nhds_hasBasis).2 _
[GOAL]
case intro.intro.refine'_2
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
⊢ ∀ (i' : Set X), i' ∈ B ∧ x ∈ i' → ∃ i, 0 < i ∧ F ⁻¹' ball (F x) i ⊆ i'
[PROOFSTEP]
rintro V ⟨hVB, hxV⟩
[GOAL]
case intro.intro.refine'_2.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
V : Set X
hVB : V ∈ B
hxV : x ∈ V
⊢ ∃ i, 0 < i ∧ F ⁻¹' ball (F x) i ⊆ V
[PROOFSTEP]
rcases hB.exists_closure_subset (hB.mem_nhds hVB hxV) with ⟨U, hUB, hxU, hUV⟩
[GOAL]
case intro.intro.refine'_2.intro.intro.intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
V : Set X
hVB : V ∈ B
hxV : x ∈ V
U : Set X
hUB : U ∈ B
hxU : x ∈ U
hUV : closure U ⊆ V
⊢ ∃ i, 0 < i ∧ F ⁻¹' ball (F x) i ⊆ V
[PROOFSTEP]
set UV : ↥s := ⟨(U, V), ⟨hUB, hVB⟩, hUV⟩
[GOAL]
case intro.intro.refine'_2.intro.intro.intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
V : Set X
hVB : V ∈ B
hxV : x ∈ V
U : Set X
hUB : U ∈ B
hxU : x ∈ U
hUV : closure U ⊆ V
UV : ↑s := { val := (U, V), property := (_ : (U, V) ∈ B ×ˢ B ∧ closure (U, V).fst ⊆ (U, V).snd) }
⊢ ∃ i, 0 < i ∧ F ⁻¹' ball (F x) i ⊆ V
[PROOFSTEP]
refine' ⟨ε UV, (ε01 UV).1, fun y (hy : dist (F y) (F x) < ε UV) => _⟩
[GOAL]
case intro.intro.refine'_2.intro.intro.intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
V : Set X
hVB : V ∈ B
hxV : x ∈ V
U : Set X
hUB : U ∈ B
hxU : x ∈ U
hUV : closure U ⊆ V
UV : ↑s := { val := (U, V), property := (_ : (U, V) ∈ B ×ˢ B ∧ closure (U, V).fst ⊆ (U, V).snd) }
y : X
hy : dist (F y) (F x) < ε UV
⊢ y ∈ V
[PROOFSTEP]
replace hy : dist (F y UV) (F x UV) < ε UV
[GOAL]
case hy
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
V : Set X
hVB : V ∈ B
hxV : x ∈ V
U : Set X
hUB : U ∈ B
hxU : x ∈ U
hUV : closure U ⊆ V
UV : ↑s := { val := (U, V), property := (_ : (U, V) ∈ B ×ˢ B ∧ closure (U, V).fst ⊆ (U, V).snd) }
y : X
hy : dist (F y) (F x) < ε UV
⊢ dist (↑(F y) UV) (↑(F x) UV) < ε UV
case intro.intro.refine'_2.intro.intro.intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
V : Set X
hVB : V ∈ B
hxV : x ∈ V
U : Set X
hUB : U ∈ B
hxU : x ∈ U
hUV : closure U ⊆ V
UV : ↑s := { val := (U, V), property := (_ : (U, V) ∈ B ×ˢ B ∧ closure (U, V).fst ⊆ (U, V).snd) }
y : X
hy : dist (↑(F y) UV) (↑(F x) UV) < ε UV
⊢ y ∈ V
[PROOFSTEP]
exact (BoundedContinuousFunction.dist_coe_le_dist _).trans_lt hy
[GOAL]
case intro.intro.refine'_2.intro.intro.intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
V : Set X
hVB : V ∈ B
hxV : x ∈ V
U : Set X
hUB : U ∈ B
hxU : x ∈ U
hUV : closure U ⊆ V
UV : ↑s := { val := (U, V), property := (_ : (U, V) ∈ B ×ˢ B ∧ closure (U, V).fst ⊆ (U, V).snd) }
y : X
hy : dist (↑(F y) UV) (↑(F x) UV) < ε UV
⊢ y ∈ V
[PROOFSTEP]
contrapose! hy
[GOAL]
case intro.intro.refine'_2.intro.intro.intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
V : Set X
hVB : V ∈ B
hxV : x ∈ V
U : Set X
hUB : U ∈ B
hxU : x ∈ U
hUV : closure U ⊆ V
UV : ↑s := { val := (U, V), property := (_ : (U, V) ∈ B ×ˢ B ∧ closure (U, V).fst ⊆ (U, V).snd) }
y : X
hy : ¬y ∈ V
⊢ ε { val := (U, V), property := (_ : (U, V) ∈ B ×ˢ B ∧ closure (U, V).fst ⊆ (U, V).snd) } ≤
    dist
      (↑((fun x =>
              { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
                map_bounded' :=
                  (_ :
                    ∃ C,
                      ∀ (x_1 y : ↑s),
                        dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                            (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                          C) })
            y)
        { val := (U, V), property := (_ : (U, V) ∈ B ×ˢ B ∧ closure (U, V).fst ⊆ (U, V).snd) })
      (↑((fun x =>
              { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
                map_bounded' :=
                  (_ :
                    ∃ C,
                      ∀ (x_1 y : ↑s),
                        dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                            (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                          C) })
            x)
        { val := (U, V), property := (_ : (U, V) ∈ B ×ˢ B ∧ closure (U, V).fst ⊆ (U, V).snd) })
[PROOFSTEP]
rw [hF, hF, hfε UV hy, hf0 UV hxU, Pi.zero_apply, dist_zero_right]
[GOAL]
case intro.intro.refine'_2.intro.intro.intro.intro
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
V : Set X
hVB : V ∈ B
hxV : x ∈ V
U : Set X
hUB : U ∈ B
hxU : x ∈ U
hUV : closure U ⊆ V
UV : ↑s := { val := (U, V), property := (_ : (U, V) ∈ B ×ˢ B ∧ closure (U, V).fst ⊆ (U, V).snd) }
y : X
hy : ¬y ∈ V
⊢ ε { val := (U, V), property := (_ : (U, V) ∈ B ×ˢ B ∧ closure (U, V).fst ⊆ (U, V).snd) } ≤ ‖(fun x => ε UV) y‖
[PROOFSTEP]
exact le_abs_self _
[GOAL]
case intro.intro.refine'_3
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
⊢ 𝓝 x ≤ comap F (𝓝 (F x))
[PROOFSTEP]
refine' (nhds_basis_closedBall.comap _).ge_iff.2 fun δ δ0 => _
[GOAL]
case intro.intro.refine'_3
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
δ : ℝ
δ0 : 0 < δ
⊢ F ⁻¹' closedBall (F x) δ ∈ 𝓝 x
[PROOFSTEP]
have h_fin : {UV : s | δ ≤ ε UV}.Finite := by simpa only [← not_lt] using hε (gt_mem_nhds δ0)
[GOAL]
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
δ : ℝ
δ0 : 0 < δ
⊢ Set.Finite {UV | δ ≤ ε UV}
[PROOFSTEP]
simpa only [← not_lt] using hε (gt_mem_nhds δ0)
[GOAL]
case intro.intro.refine'_3
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
δ : ℝ
δ0 : 0 < δ
h_fin : Set.Finite {UV | δ ≤ ε UV}
⊢ F ⁻¹' closedBall (F x) δ ∈ 𝓝 x
[PROOFSTEP]
have : ∀ᶠ y in 𝓝 x, ∀ UV, δ ≤ ε UV → dist (F y UV) (F x UV) ≤ δ :=
  by
  refine' (eventually_all_finite h_fin).2 fun UV _ => _
  exact (f UV).continuous.tendsto x (closedBall_mem_nhds _ δ0)
[GOAL]
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
δ : ℝ
δ0 : 0 < δ
h_fin : Set.Finite {UV | δ ≤ ε UV}
⊢ ∀ᶠ (y : X) in 𝓝 x, ∀ (UV : ↑s), δ ≤ ε UV → dist (↑(F y) UV) (↑(F x) UV) ≤ δ
[PROOFSTEP]
refine' (eventually_all_finite h_fin).2 fun UV _ => _
[GOAL]
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝² : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝¹ : Encodable ↑s
this✝ : TopologicalSpace ↑s := ⊥
this : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
δ : ℝ
δ0 : 0 < δ
h_fin : Set.Finite {UV | δ ≤ ε UV}
UV : ↑s
x✝ : UV ∈ {UV | δ ≤ ε UV}
⊢ ∀ᶠ (x_1 : X) in 𝓝 x, dist (↑(F x_1) UV) (↑(F x) UV) ≤ δ
[PROOFSTEP]
exact (f UV).continuous.tendsto x (closedBall_mem_nhds _ δ0)
[GOAL]
case intro.intro.refine'_3
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝³ : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝² : Encodable ↑s
this✝¹ : TopologicalSpace ↑s := ⊥
this✝ : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
δ : ℝ
δ0 : 0 < δ
h_fin : Set.Finite {UV | δ ≤ ε UV}
this : ∀ᶠ (y : X) in 𝓝 x, ∀ (UV : ↑s), δ ≤ ε UV → dist (↑(F y) UV) (↑(F x) UV) ≤ δ
⊢ F ⁻¹' closedBall (F x) δ ∈ 𝓝 x
[PROOFSTEP]
refine' this.mono fun y hy => (BoundedContinuousFunction.dist_le δ0.le).2 fun UV => _
[GOAL]
case intro.intro.refine'_3
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝³ : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝² : Encodable ↑s
this✝¹ : TopologicalSpace ↑s := ⊥
this✝ : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
δ : ℝ
δ0 : 0 < δ
h_fin : Set.Finite {UV | δ ≤ ε UV}
this : ∀ᶠ (y : X) in 𝓝 x, ∀ (UV : ↑s), δ ≤ ε UV → dist (↑(F y) UV) (↑(F x) UV) ≤ δ
y : X
hy : ∀ (UV : ↑s), δ ≤ ε UV → dist (↑(F y) UV) (↑(F x) UV) ≤ δ
UV : ↑s
⊢ dist (↑(F y) UV) (↑(F x) UV) ≤ δ
[PROOFSTEP]
cases' le_total δ (ε UV) with hle hle
[GOAL]
case intro.intro.refine'_3.inl
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝³ : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝² : Encodable ↑s
this✝¹ : TopologicalSpace ↑s := ⊥
this✝ : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
δ : ℝ
δ0 : 0 < δ
h_fin : Set.Finite {UV | δ ≤ ε UV}
this : ∀ᶠ (y : X) in 𝓝 x, ∀ (UV : ↑s), δ ≤ ε UV → dist (↑(F y) UV) (↑(F x) UV) ≤ δ
y : X
hy : ∀ (UV : ↑s), δ ≤ ε UV → dist (↑(F y) UV) (↑(F x) UV) ≤ δ
UV : ↑s
hle : δ ≤ ε UV
⊢ dist (↑(F y) UV) (↑(F x) UV) ≤ δ
case intro.intro.refine'_3.inr
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝³ : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝² : Encodable ↑s
this✝¹ : TopologicalSpace ↑s := ⊥
this✝ : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
δ : ℝ
δ0 : 0 < δ
h_fin : Set.Finite {UV | δ ≤ ε UV}
this : ∀ᶠ (y : X) in 𝓝 x, ∀ (UV : ↑s), δ ≤ ε UV → dist (↑(F y) UV) (↑(F x) UV) ≤ δ
y : X
hy : ∀ (UV : ↑s), δ ≤ ε UV → dist (↑(F y) UV) (↑(F x) UV) ≤ δ
UV : ↑s
hle : ε UV ≤ δ
⊢ dist (↑(F y) UV) (↑(F x) UV) ≤ δ
[PROOFSTEP]
exacts [hy _ hle, (Real.dist_le_of_mem_Icc (hf0ε _ _) (hf0ε _ _)).trans (by rwa [sub_zero])]
[GOAL]
ι : Type u_1
X : Type u_2
Y : Type u_3
π : ι → Type u_4
inst✝⁵ : TopologicalSpace X
inst✝⁴ : TopologicalSpace Y
inst✝³ : Finite ι
inst✝² : (i : ι) → TopologicalSpace (π i)
inst✝¹ : T3Space X
inst✝ : SecondCountableTopology X
this✝³ : NormalSpace X
B : Set (Set X)
hBc : Set.Countable B
hB : IsTopologicalBasis B
s : Set (Set X × Set X) := {UV | UV ∈ B ×ˢ B ∧ closure UV.fst ⊆ UV.snd}
this✝² : Encodable ↑s
this✝¹ : TopologicalSpace ↑s := ⊥
this✝ : DiscreteTopology ↑s
hd : ∀ (UV : ↑s), Disjoint (closure (↑UV).fst) (↑UV).sndᶜ
ε : ↑s → ℝ
ε01 : ∀ (UV : ↑s), ε UV ∈ Ioc 0 1
hε : Tendsto ε cofinite (𝓝 0)
f : ↑s → C(X, ℝ)
hf0 : ∀ (UV : ↑s), EqOn (↑(f UV)) 0 (↑UV).fst
hfε : ∀ (UV : ↑s), EqOn (↑(f UV)) (fun x => ε UV) (↑UV).sndᶜ
hf0ε : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 (ε UV)
hf01 : ∀ (UV : ↑s) (x : X), ↑(f UV) x ∈ Icc 0 1
F : X → ↑s →ᵇ ℝ :=
  fun x =>
    { toContinuousMap := ContinuousMap.mk fun UV => ↑(f UV) x,
      map_bounded' :=
        (_ :
          ∃ C,
            ∀ (x_1 y : ↑s),
              dist (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) x_1)
                  (ContinuousMap.toFun (ContinuousMap.mk fun UV => ↑(f UV) x) y) ≤
                C) }
hF : ∀ (x : X) (UV : ↑s), ↑(F x) UV = ↑(f UV) x
x : X
δ : ℝ
δ0 : 0 < δ
h_fin : Set.Finite {UV | δ ≤ ε UV}
this : ∀ᶠ (y : X) in 𝓝 x, ∀ (UV : ↑s), δ ≤ ε UV → dist (↑(F y) UV) (↑(F x) UV) ≤ δ
y : X
hy : ∀ (UV : ↑s), δ ≤ ε UV → dist (↑(F y) UV) (↑(F x) UV) ≤ δ
UV : ↑s
hle : ε UV ≤ δ
⊢ ε UV - 0 ≤ δ
[PROOFSTEP]
rwa [sub_zero]