File: Mathlib/Topology/MetricSpace/PartitionOfUnity.lean

full_name: EMetric.eventually_nhds_zero_forall_closedBall_subset
code: /-- Let `K : ι → Set X` be a locally finite family of closed sets in an emetric space. Let
`U : ι → Set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then for any point
`x : X`, for sufficiently small `r : ℝ≥0∞` and for `y` sufficiently close to `x`, for all `i`, if
`y ∈ K i`, then `EMetric.closedBall y r ⊆ U i`. -/
theorem eventually_nhds_zero_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
    (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) (x : X) :
    ∀ᶠ p : ℝ≥0∞ × X in 𝓝 0 ×ˢ 𝓝 x, ∀ i, p.2 ∈ K i → closedBall p.2 p.1 ⊆ U i 
goal: ∀ᶠ (p : ℝ≥0∞ × X) in 𝓝 0 ×ˢ 𝓝 x, ∀ (i : ι), p.2 ∈ K i → EMetric.closedBall p.2 p.1 ⊆ U i

full_name: EMetric.exists_forall_closedBall_subset_aux₁
code: theorem exists_forall_closedBall_subset_aux₁ (hK : ∀ i, IsClosed (K i)) (hU : ∀ i, IsOpen (U i))
    (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) (x : X) :
    ∃ r : ℝ, ∀ᶠ y in 𝓝 x,
      r ∈ Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (_ : y ∈ K i), { r | closedBall y r ⊆ U i } 
goal: ∃ r, ∀ᶠ (y : X) in 𝓝 x, r ∈ Set.Ioi 0 ∩ ENNReal.ofReal ⁻¹' ⋂ i, ⋂ (_ : y ∈ K i), {r | EMetric.closedBall y r ⊆ U i}

full_name: EMetric.exists_forall_closedBall_subset_aux₂
code: theorem exists_forall_closedBall_subset_aux₂ (y : X) :
    Convex ℝ
      (Ioi (0 : ℝ) ∩ ENNReal.ofReal ⁻¹' ⋂ (i) (_ : y ∈ K i), { r | closedBall y r ⊆ U i }) 
goal: Convex ℝ (Set.Ioi 0 ∩ ENNReal.ofReal ⁻¹' ⋂ i, ⋂ (_ : y ∈ K i), {r | EMetric.closedBall y r ⊆ U i})

full_name: EMetric.exists_continuous_real_forall_closedBall_subset
code: /-- Let `X` be an extended metric space. Let `K : ι → Set X` be a locally finite family of closed
sets, let `U : ι → Set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
exists a positive continuous function `δ : C(X, ℝ)` such that for any `i` and `x ∈ K i`,
we have `EMetric.closedBall x (ENNReal.ofReal (δ x)) ⊆ U i`. -/
theorem exists_continuous_real_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
    (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
    ∃ δ : C(X, ℝ), (∀ x, 0 < δ x) ∧
      ∀ (i), ∀ x ∈ K i, closedBall x (ENNReal.ofReal <| δ x) ⊆ U i 
goal: ∃ δ, (∀ (x : X), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, EMetric.closedBall x (ENNReal.ofReal (δ x)) ⊆ U i

full_name: EMetric.exists_continuous_nnreal_forall_closedBall_subset
code: /-- Let `X` be an extended metric space. Let `K : ι → Set X` be a locally finite family of closed
sets, let `U : ι → Set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
exists a positive continuous function `δ : C(X, ℝ≥0)` such that for any `i` and `x ∈ K i`,
we have `EMetric.closedBall x (δ x) ⊆ U i`. -/
theorem exists_continuous_nnreal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
    (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
    ∃ δ : C(X, ℝ≥0), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i 
goal: ∃ δ, (∀ (x : X), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, EMetric.closedBall x ↑(δ x) ⊆ U i

full_name: EMetric.exists_continuous_eNNReal_forall_closedBall_subset
code: /-- Let `X` be an extended metric space. Let `K : ι → Set X` be a locally finite family of closed
sets, let `U : ι → Set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there
exists a positive continuous function `δ : C(X, ℝ≥0∞)` such that for any `i` and `x ∈ K i`,
we have `EMetric.closedBall x (δ x) ⊆ U i`. -/
theorem exists_continuous_eNNReal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
    (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
    ∃ δ : C(X, ℝ≥0∞), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i 
goal: ∃ δ, (∀ (x : X), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, EMetric.closedBall x (δ x) ⊆ U i

full_name: Metric.exists_continuous_nnreal_forall_closedBall_subset
code: /-- Let `X` be a metric space. Let `K : ι → Set X` be a locally finite family of closed sets, let
`U : ι → Set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there exists a
positive continuous function `δ : C(X, ℝ≥0)` such that for any `i` and `x ∈ K i`, we have
`Metric.closedBall x (δ x) ⊆ U i`. -/
theorem exists_continuous_nnreal_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
    (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
    ∃ δ : C(X, ℝ≥0), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i 
goal: ∃ δ, (∀ (x : X), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, Metric.closedBall x ↑(δ x) ⊆ U i

full_name: Metric.exists_continuous_real_forall_closedBall_subset
code: /-- Let `X` be a metric space. Let `K : ι → Set X` be a locally finite family of closed sets, let
`U : ι → Set X` be a family of open sets such that `K i ⊆ U i` for all `i`. Then there exists a
positive continuous function `δ : C(X, ℝ)` such that for any `i` and `x ∈ K i`, we have
`Metric.closedBall x (δ x) ⊆ U i`. -/
theorem exists_continuous_real_forall_closedBall_subset (hK : ∀ i, IsClosed (K i))
    (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i) (hfin : LocallyFinite K) :
    ∃ δ : C(X, ℝ), (∀ x, 0 < δ x) ∧ ∀ (i), ∀ x ∈ K i, closedBall x (δ x) ⊆ U i 
goal: ∃ δ, (∀ (x : X), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, Metric.closedBall x (δ x) ⊆ U i