fact
stringlengths 6
3.84k
| type
stringclasses 11
values | library
stringclasses 32
values | imports
listlengths 1
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| filename
stringlengths 20
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| symbolic_name
stringlengths 1
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| docstring
stringlengths 7
20k
⌀ |
|---|---|---|---|---|---|---|
mem_closedBall_self (h : 0 ≤ ε) : x ∈ closedBall x ε := by
rwa [mem_closedBall, dist_self]
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
mem_closedBall_self
| null |
nonempty_closedBall : (closedBall x ε).Nonempty ↔ 0 ≤ ε :=
⟨fun ⟨_x, hx⟩ => dist_nonneg.trans hx, fun h => ⟨x, mem_closedBall_self h⟩⟩
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
nonempty_closedBall
| null |
closedBall_eq_empty : closedBall x ε = ∅ ↔ ε < 0 := by
rw [← not_nonempty_iff_eq_empty, nonempty_closedBall, not_le]
@[simp] alias ⟨_, closedBall_of_neg⟩ := closedBall_eq_empty
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
closedBall_eq_empty
| null |
closedBall_eq_sphere_of_nonpos (hε : ε ≤ 0) : closedBall x ε = sphere x ε :=
Set.ext fun _ => (hε.trans dist_nonneg).ge_iff_eq'
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
closedBall_eq_sphere_of_nonpos
|
Closed balls and spheres coincide when the radius is non-positive
|
ball_subset_closedBall : ball x ε ⊆ closedBall x ε := fun _y hy =>
mem_closedBall.2 (le_of_lt hy)
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
ball_subset_closedBall
| null |
sphere_subset_closedBall : sphere x ε ⊆ closedBall x ε := fun _ => le_of_eq
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
sphere_subset_closedBall
| null |
sphere_subset_ball {r R : ℝ} (h : r < R) : sphere x r ⊆ ball x R := fun _x hx ↦
(mem_sphere.1 hx).trans_lt h
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
sphere_subset_ball
| null |
closedBall_disjoint_ball (h : δ + ε ≤ dist x y) : Disjoint (closedBall x δ) (ball y ε) :=
Set.disjoint_left.mpr fun _a ha1 ha2 =>
(h.trans <| dist_triangle_left _ _ _).not_gt <| add_lt_add_of_le_of_lt ha1 ha2
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
closedBall_disjoint_ball
| null |
ball_disjoint_closedBall (h : δ + ε ≤ dist x y) : Disjoint (ball x δ) (closedBall y ε) :=
(closedBall_disjoint_ball <| by rwa [add_comm, dist_comm]).symm
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
ball_disjoint_closedBall
| null |
ball_disjoint_ball (h : δ + ε ≤ dist x y) : Disjoint (ball x δ) (ball y ε) :=
(closedBall_disjoint_ball h).mono_left ball_subset_closedBall
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
ball_disjoint_ball
| null |
closedBall_disjoint_closedBall (h : δ + ε < dist x y) :
Disjoint (closedBall x δ) (closedBall y ε) :=
Set.disjoint_left.mpr fun _a ha1 ha2 =>
h.not_ge <| (dist_triangle_left _ _ _).trans <| add_le_add ha1 ha2
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
closedBall_disjoint_closedBall
| null |
sphere_disjoint_ball : Disjoint (sphere x ε) (ball x ε) :=
Set.disjoint_left.mpr fun _y hy₁ hy₂ => absurd hy₁ <| ne_of_lt hy₂
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
sphere_disjoint_ball
| null |
ball_union_sphere : ball x ε ∪ sphere x ε = closedBall x ε :=
Set.ext fun _y => (@le_iff_lt_or_eq ℝ _ _ _).symm
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
ball_union_sphere
| null |
sphere_union_ball : sphere x ε ∪ ball x ε = closedBall x ε := by
rw [union_comm, ball_union_sphere]
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
sphere_union_ball
| null |
closedBall_diff_sphere : closedBall x ε \ sphere x ε = ball x ε := by
rw [← ball_union_sphere, Set.union_diff_cancel_right sphere_disjoint_ball.symm.le_bot]
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
closedBall_diff_sphere
| null |
closedBall_diff_ball : closedBall x ε \ ball x ε = sphere x ε := by
rw [← ball_union_sphere, Set.union_diff_cancel_left sphere_disjoint_ball.symm.le_bot]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
closedBall_diff_ball
| null |
mem_ball_comm : x ∈ ball y ε ↔ y ∈ ball x ε := by rw [mem_ball', mem_ball]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
mem_ball_comm
| null |
mem_closedBall_comm : x ∈ closedBall y ε ↔ y ∈ closedBall x ε := by
rw [mem_closedBall', mem_closedBall]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
mem_closedBall_comm
| null |
mem_sphere_comm : x ∈ sphere y ε ↔ y ∈ sphere x ε := by rw [mem_sphere', mem_sphere]
@[gcongr]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
mem_sphere_comm
| null |
ball_subset_ball (h : ε₁ ≤ ε₂) : ball x ε₁ ⊆ ball x ε₂ := fun _y yx =>
lt_of_lt_of_le (mem_ball.1 yx) h
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
ball_subset_ball
| null |
closedBall_eq_bInter_ball : closedBall x ε = ⋂ δ > ε, ball x δ := by
ext y; rw [mem_closedBall, ← forall_gt_iff_le, mem_iInter₂]; rfl
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
closedBall_eq_bInter_ball
| null |
ball_subset_ball' (h : ε₁ + dist x y ≤ ε₂) : ball x ε₁ ⊆ ball y ε₂ := fun z hz =>
calc
dist z y ≤ dist z x + dist x y := dist_triangle _ _ _
_ < ε₁ + dist x y := add_lt_add_right (mem_ball.1 hz) _
_ ≤ ε₂ := h
@[gcongr]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
ball_subset_ball'
| null |
closedBall_subset_closedBall (h : ε₁ ≤ ε₂) : closedBall x ε₁ ⊆ closedBall x ε₂ :=
fun _y (yx : _ ≤ ε₁) => le_trans yx h
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
closedBall_subset_closedBall
| null |
closedBall_subset_closedBall' (h : ε₁ + dist x y ≤ ε₂) :
closedBall x ε₁ ⊆ closedBall y ε₂ := fun z hz =>
calc
dist z y ≤ dist z x + dist x y := dist_triangle _ _ _
_ ≤ ε₁ + dist x y := add_le_add_right (mem_closedBall.1 hz) _
_ ≤ ε₂ := h
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
closedBall_subset_closedBall'
| null |
closedBall_subset_ball (h : ε₁ < ε₂) : closedBall x ε₁ ⊆ ball x ε₂ :=
fun y (yh : dist y x ≤ ε₁) => lt_of_le_of_lt yh h
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
closedBall_subset_ball
| null |
closedBall_subset_ball' (h : ε₁ + dist x y < ε₂) :
closedBall x ε₁ ⊆ ball y ε₂ := fun z hz =>
calc
dist z y ≤ dist z x + dist x y := dist_triangle _ _ _
_ ≤ ε₁ + dist x y := add_le_add_right (mem_closedBall.1 hz) _
_ < ε₂ := h
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
closedBall_subset_ball'
| null |
dist_le_add_of_nonempty_closedBall_inter_closedBall
(h : (closedBall x ε₁ ∩ closedBall y ε₂).Nonempty) : dist x y ≤ ε₁ + ε₂ :=
let ⟨z, hz⟩ := h
calc
dist x y ≤ dist z x + dist z y := dist_triangle_left _ _ _
_ ≤ ε₁ + ε₂ := add_le_add hz.1 hz.2
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_le_add_of_nonempty_closedBall_inter_closedBall
| null |
dist_lt_add_of_nonempty_closedBall_inter_ball (h : (closedBall x ε₁ ∩ ball y ε₂).Nonempty) :
dist x y < ε₁ + ε₂ :=
let ⟨z, hz⟩ := h
calc
dist x y ≤ dist z x + dist z y := dist_triangle_left _ _ _
_ < ε₁ + ε₂ := add_lt_add_of_le_of_lt hz.1 hz.2
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_lt_add_of_nonempty_closedBall_inter_ball
| null |
dist_lt_add_of_nonempty_ball_inter_closedBall (h : (ball x ε₁ ∩ closedBall y ε₂).Nonempty) :
dist x y < ε₁ + ε₂ := by
rw [inter_comm] at h
rw [add_comm, dist_comm]
exact dist_lt_add_of_nonempty_closedBall_inter_ball h
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_lt_add_of_nonempty_ball_inter_closedBall
| null |
dist_lt_add_of_nonempty_ball_inter_ball (h : (ball x ε₁ ∩ ball y ε₂).Nonempty) :
dist x y < ε₁ + ε₂ :=
dist_lt_add_of_nonempty_closedBall_inter_ball <|
h.mono (inter_subset_inter ball_subset_closedBall Subset.rfl)
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_lt_add_of_nonempty_ball_inter_ball
| null |
iUnion_closedBall_nat (x : α) : ⋃ n : ℕ, closedBall x n = univ :=
iUnion_eq_univ_iff.2 fun y => exists_nat_ge (dist y x)
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
iUnion_closedBall_nat
| null |
iUnion_inter_closedBall_nat (s : Set α) (x : α) : ⋃ n : ℕ, s ∩ closedBall x n = s := by
rw [← inter_iUnion, iUnion_closedBall_nat, inter_univ]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
iUnion_inter_closedBall_nat
| null |
ball_subset (h : dist x y ≤ ε₂ - ε₁) : ball x ε₁ ⊆ ball y ε₂ := fun z zx => by
rw [← add_sub_cancel ε₁ ε₂]
exact lt_of_le_of_lt (dist_triangle z x y) (add_lt_add_of_lt_of_le zx h)
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
ball_subset
| null |
ball_half_subset (y) (h : y ∈ ball x (ε / 2)) : ball y (ε / 2) ⊆ ball x ε :=
ball_subset <| by rw [sub_self_div_two]; exact le_of_lt h
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
ball_half_subset
| null |
exists_ball_subset_ball (h : y ∈ ball x ε) : ∃ ε' > 0, ball y ε' ⊆ ball x ε :=
⟨_, sub_pos.2 h, ball_subset <| by rw [sub_sub_self]⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
exists_ball_subset_ball
| null |
forall_of_forall_mem_closedBall (p : α → Prop) (x : α)
(H : ∃ᶠ R : ℝ in atTop, ∀ y ∈ closedBall x R, p y) (y : α) : p y := by
obtain ⟨R, hR, h⟩ : ∃ R ≥ dist y x, ∀ z : α, z ∈ closedBall x R → p z :=
frequently_iff.1 H (Ici_mem_atTop (dist y x))
exact h _ hR
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
forall_of_forall_mem_closedBall
|
If a property holds for all points in closed balls of arbitrarily large radii, then it holds for
all points.
|
forall_of_forall_mem_ball (p : α → Prop) (x : α)
(H : ∃ᶠ R : ℝ in atTop, ∀ y ∈ ball x R, p y) (y : α) : p y := by
obtain ⟨R, hR, h⟩ : ∃ R > dist y x, ∀ z : α, z ∈ ball x R → p z :=
frequently_iff.1 H (Ioi_mem_atTop (dist y x))
exact h _ hR
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
forall_of_forall_mem_ball
|
If a property holds for all points in balls of arbitrarily large radii, then it holds for all
points.
|
isBounded_iff {s : Set α} :
IsBounded s ↔ ∃ C : ℝ, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C := by
rw [isBounded_def, ← Filter.mem_sets, @PseudoMetricSpace.cobounded_sets α, mem_setOf_eq,
compl_compl]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
isBounded_iff
| null |
isBounded_iff_eventually {s : Set α} :
IsBounded s ↔ ∀ᶠ C in atTop, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C :=
isBounded_iff.trans
⟨fun ⟨C, h⟩ => eventually_atTop.2 ⟨C, fun _C' hC' _x hx _y hy => (h hx hy).trans hC'⟩,
Eventually.exists⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
isBounded_iff_eventually
| null |
isBounded_iff_exists_ge {s : Set α} (c : ℝ) :
IsBounded s ↔ ∃ C, c ≤ C ∧ ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → dist x y ≤ C :=
⟨fun h => ((eventually_ge_atTop c).and (isBounded_iff_eventually.1 h)).exists, fun h =>
isBounded_iff.2 <| h.imp fun _ => And.right⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
isBounded_iff_exists_ge
| null |
isBounded_iff_nndist {s : Set α} :
IsBounded s ↔ ∃ C : ℝ≥0, ∀ ⦃x⦄, x ∈ s → ∀ ⦃y⦄, y ∈ s → nndist x y ≤ C := by
simp only [isBounded_iff_exists_ge 0, NNReal.exists, ← NNReal.coe_le_coe, ← dist_nndist,
NNReal.coe_mk, exists_prop]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
isBounded_iff_nndist
| null |
toUniformSpace_eq :
‹PseudoMetricSpace α›.toUniformSpace = .ofDist dist dist_self dist_comm dist_triangle :=
UniformSpace.ext PseudoMetricSpace.uniformity_dist
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
toUniformSpace_eq
| null |
uniformity_basis_dist :
(𝓤 α).HasBasis (fun ε : ℝ => 0 < ε) fun ε => { p : α × α | dist p.1 p.2 < ε } := by
rw [toUniformSpace_eq]
exact UniformSpace.hasBasis_ofFun (exists_gt _) _ _ _ _ _
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
uniformity_basis_dist
| null |
protected mk_uniformity_basis {β : Type*} {p : β → Prop} {f : β → ℝ}
(hf₀ : ∀ i, p i → 0 < f i) (hf : ∀ ⦃ε⦄, 0 < ε → ∃ i, p i ∧ f i ≤ ε) :
(𝓤 α).HasBasis p fun i => { p : α × α | dist p.1 p.2 < f i } := by
refine ⟨fun s => uniformity_basis_dist.mem_iff.trans ?_⟩
constructor
· rintro ⟨ε, ε₀, hε⟩
rcases hf ε₀ with ⟨i, hi, H⟩
exact ⟨i, hi, fun x (hx : _ < _) => hε <| lt_of_lt_of_le hx H⟩
· exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, H⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
mk_uniformity_basis
|
Given `f : β → ℝ`, if `f` sends `{i | p i}` to a set of positive numbers
accumulating to zero, then `f i`-neighborhoods of the diagonal form a basis of `𝓤 α`.
For specific bases see `uniformity_basis_dist`, `uniformity_basis_dist_inv_nat_succ`,
and `uniformity_basis_dist_inv_nat_pos`.
|
uniformity_basis_dist_rat :
(𝓤 α).HasBasis (fun r : ℚ => 0 < r) fun r => { p : α × α | dist p.1 p.2 < r } :=
Metric.mk_uniformity_basis (fun _ => Rat.cast_pos.2) fun _ε hε =>
let ⟨r, hr0, hrε⟩ := exists_rat_btwn hε
⟨r, Rat.cast_pos.1 hr0, hrε.le⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
uniformity_basis_dist_rat
| null |
uniformity_basis_dist_inv_nat_succ :
(𝓤 α).HasBasis (fun _ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 < 1 / (↑n + 1) } :=
Metric.mk_uniformity_basis (fun n _ => div_pos zero_lt_one <| Nat.cast_add_one_pos n) fun _ε ε0 =>
(exists_nat_one_div_lt ε0).imp fun _n hn => ⟨trivial, le_of_lt hn⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
uniformity_basis_dist_inv_nat_succ
| null |
uniformity_basis_dist_inv_nat_pos :
(𝓤 α).HasBasis (fun n : ℕ => 0 < n) fun n : ℕ => { p : α × α | dist p.1 p.2 < 1 / ↑n } :=
Metric.mk_uniformity_basis (fun _ hn => div_pos zero_lt_one <| Nat.cast_pos.2 hn) fun _ ε0 =>
let ⟨n, hn⟩ := exists_nat_one_div_lt ε0
⟨n + 1, Nat.succ_pos n, mod_cast hn.le⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
uniformity_basis_dist_inv_nat_pos
| null |
uniformity_basis_dist_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
(𝓤 α).HasBasis (fun _ : ℕ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 < r ^ n } :=
Metric.mk_uniformity_basis (fun _ _ => pow_pos h0 _) fun _ε ε0 =>
let ⟨n, hn⟩ := exists_pow_lt_of_lt_one ε0 h1
⟨n, trivial, hn.le⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
uniformity_basis_dist_pow
| null |
uniformity_basis_dist_lt {R : ℝ} (hR : 0 < R) :
(𝓤 α).HasBasis (fun r : ℝ => 0 < r ∧ r < R) fun r => { p : α × α | dist p.1 p.2 < r } :=
Metric.mk_uniformity_basis (fun _ => And.left) fun r hr =>
⟨min r (R / 2), ⟨lt_min hr (half_pos hR), min_lt_iff.2 <| Or.inr (half_lt_self hR)⟩,
min_le_left _ _⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
uniformity_basis_dist_lt
| null |
protected mk_uniformity_basis_le {β : Type*} {p : β → Prop} {f : β → ℝ}
(hf₀ : ∀ x, p x → 0 < f x) (hf : ∀ ε, 0 < ε → ∃ x, p x ∧ f x ≤ ε) :
(𝓤 α).HasBasis p fun x => { p : α × α | dist p.1 p.2 ≤ f x } := by
refine ⟨fun s => uniformity_basis_dist.mem_iff.trans ?_⟩
constructor
· rintro ⟨ε, ε₀, hε⟩
rcases exists_between ε₀ with ⟨ε', hε'⟩
rcases hf ε' hε'.1 with ⟨i, hi, H⟩
exact ⟨i, hi, fun x (hx : _ ≤ _) => hε <| lt_of_le_of_lt (le_trans hx H) hε'.2⟩
· exact fun ⟨i, hi, H⟩ => ⟨f i, hf₀ i hi, fun x (hx : _ < _) => H (mem_setOf.2 hx.le)⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
mk_uniformity_basis_le
|
Given `f : β → ℝ`, if `f` sends `{i | p i}` to a set of positive numbers
accumulating to zero, then closed neighborhoods of the diagonal of sizes `{f i | p i}`
form a basis of `𝓤 α`.
Currently we have only one specific basis `uniformity_basis_dist_le` based on this constructor.
More can be easily added if needed in the future.
|
uniformity_basis_dist_le :
(𝓤 α).HasBasis ((0 : ℝ) < ·) fun ε => { p : α × α | dist p.1 p.2 ≤ ε } :=
Metric.mk_uniformity_basis_le (fun _ => id) fun ε ε₀ => ⟨ε, ε₀, le_refl ε⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
uniformity_basis_dist_le
|
Constant size closed neighborhoods of the diagonal form a basis
of the uniformity filter.
|
uniformity_basis_dist_le_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
(𝓤 α).HasBasis (fun _ : ℕ => True) fun n : ℕ => { p : α × α | dist p.1 p.2 ≤ r ^ n } :=
Metric.mk_uniformity_basis_le (fun _ _ => pow_pos h0 _) fun _ε ε0 =>
let ⟨n, hn⟩ := exists_pow_lt_of_lt_one ε0 h1
⟨n, trivial, hn.le⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
uniformity_basis_dist_le_pow
| null |
mem_uniformity_dist {s : Set (α × α)} :
s ∈ 𝓤 α ↔ ∃ ε > 0, ∀ ⦃a b : α⦄, dist a b < ε → (a, b) ∈ s :=
uniformity_basis_dist.mem_uniformity_iff
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
mem_uniformity_dist
| null |
dist_mem_uniformity {ε : ℝ} (ε0 : 0 < ε) : { p : α × α | dist p.1 p.2 < ε } ∈ 𝓤 α :=
mem_uniformity_dist.2 ⟨ε, ε0, fun _ _ ↦ id⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
dist_mem_uniformity
|
A constant size neighborhood of the diagonal is an entourage.
|
uniformContinuous_iff [PseudoMetricSpace β] {f : α → β} :
UniformContinuous f ↔ ∀ ε > 0, ∃ δ > 0, ∀ ⦃a b : α⦄, dist a b < δ → dist (f a) (f b) < ε :=
uniformity_basis_dist.uniformContinuous_iff uniformity_basis_dist
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
uniformContinuous_iff
| null |
uniformContinuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} :
UniformContinuousOn f s ↔
∀ ε > 0, ∃ δ > 0, ∀ x ∈ s, ∀ y ∈ s, dist x y < δ → dist (f x) (f y) < ε :=
Metric.uniformity_basis_dist.uniformContinuousOn_iff Metric.uniformity_basis_dist
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
uniformContinuousOn_iff
| null |
uniformContinuousOn_iff_le [PseudoMetricSpace β] {f : α → β} {s : Set α} :
UniformContinuousOn f s ↔
∀ ε > 0, ∃ δ > 0, ∀ x ∈ s, ∀ y ∈ s, dist x y ≤ δ → dist (f x) (f y) ≤ ε :=
Metric.uniformity_basis_dist_le.uniformContinuousOn_iff Metric.uniformity_basis_dist_le
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
uniformContinuousOn_iff_le
| null |
nhds_basis_ball : (𝓝 x).HasBasis (0 < ·) (ball x) :=
nhds_basis_uniformity uniformity_basis_dist
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
nhds_basis_ball
| null |
mem_nhds_iff : s ∈ 𝓝 x ↔ ∃ ε > 0, ball x ε ⊆ s :=
nhds_basis_ball.mem_iff
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
mem_nhds_iff
| null |
eventually_nhds_iff {p : α → Prop} :
(∀ᶠ y in 𝓝 x, p y) ↔ ∃ ε > 0, ∀ ⦃y⦄, dist y x < ε → p y :=
mem_nhds_iff
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
eventually_nhds_iff
| null |
eventually_nhds_iff_ball {p : α → Prop} :
(∀ᶠ y in 𝓝 x, p y) ↔ ∃ ε > 0, ∀ y ∈ ball x ε, p y :=
mem_nhds_iff
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
eventually_nhds_iff_ball
| null |
eventually_nhds_prod_iff {f : Filter ι} {x₀ : α} {p : α × ι → Prop} :
(∀ᶠ x in 𝓝 x₀ ×ˢ f, p x) ↔ ∃ ε > (0 : ℝ), ∃ pa : ι → Prop, (∀ᶠ i in f, pa i) ∧
∀ ⦃x⦄, dist x x₀ < ε → ∀ ⦃i⦄, pa i → p (x, i) := by
refine (nhds_basis_ball.prod f.basis_sets).eventually_iff.trans ?_
simp only [Prod.exists, forall_prod_set, id, mem_ball, and_assoc, exists_and_left]
rfl
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
eventually_nhds_prod_iff
|
A version of `Filter.eventually_prod_iff` where the first filter consists of neighborhoods
in a pseudo-metric space.
|
eventually_prod_nhds_iff {f : Filter ι} {x₀ : α} {p : ι × α → Prop} :
(∀ᶠ x in f ×ˢ 𝓝 x₀, p x) ↔ ∃ pa : ι → Prop, (∀ᶠ i in f, pa i) ∧
∃ ε > 0, ∀ ⦃i⦄, pa i → ∀ ⦃x⦄, dist x x₀ < ε → p (i, x) := by
rw [eventually_swap_iff, Metric.eventually_nhds_prod_iff]
constructor <;>
· rintro ⟨a1, a2, a3, a4, a5⟩
exact ⟨a3, a4, a1, a2, fun _ b1 b2 b3 => a5 b3 b1⟩
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
eventually_prod_nhds_iff
|
A version of `Filter.eventually_prod_iff` where the second filter consists of neighborhoods
in a pseudo-metric space.
|
nhds_basis_closedBall : (𝓝 x).HasBasis (fun ε : ℝ => 0 < ε) (closedBall x) :=
nhds_basis_uniformity uniformity_basis_dist_le
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
nhds_basis_closedBall
| null |
nhds_basis_ball_inv_nat_succ :
(𝓝 x).HasBasis (fun _ => True) fun n : ℕ => ball x (1 / (↑n + 1)) :=
nhds_basis_uniformity uniformity_basis_dist_inv_nat_succ
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
nhds_basis_ball_inv_nat_succ
| null |
nhds_basis_ball_inv_nat_pos :
(𝓝 x).HasBasis (fun n => 0 < n) fun n : ℕ => ball x (1 / ↑n) :=
nhds_basis_uniformity uniformity_basis_dist_inv_nat_pos
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
nhds_basis_ball_inv_nat_pos
| null |
nhds_basis_ball_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
(𝓝 x).HasBasis (fun _ => True) fun n : ℕ => ball x (r ^ n) :=
nhds_basis_uniformity (uniformity_basis_dist_pow h0 h1)
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
nhds_basis_ball_pow
| null |
nhds_basis_closedBall_pow {r : ℝ} (h0 : 0 < r) (h1 : r < 1) :
(𝓝 x).HasBasis (fun _ => True) fun n : ℕ => closedBall x (r ^ n) :=
nhds_basis_uniformity (uniformity_basis_dist_le_pow h0 h1)
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
nhds_basis_closedBall_pow
| null |
isOpen_iff : IsOpen s ↔ ∀ x ∈ s, ∃ ε > 0, ball x ε ⊆ s := by
simp only [isOpen_iff_mem_nhds, mem_nhds_iff]
@[simp] theorem isOpen_ball : IsOpen (ball x ε) :=
isOpen_iff.2 fun _ => exists_ball_subset_ball
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
isOpen_iff
| null |
ball_mem_nhds (x : α) {ε : ℝ} (ε0 : 0 < ε) : ball x ε ∈ 𝓝 x :=
isOpen_ball.mem_nhds (mem_ball_self ε0)
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
ball_mem_nhds
| null |
closedBall_mem_nhds (x : α) {ε : ℝ} (ε0 : 0 < ε) : closedBall x ε ∈ 𝓝 x :=
mem_of_superset (ball_mem_nhds x ε0) ball_subset_closedBall
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
closedBall_mem_nhds
| null |
closedBall_mem_nhds_of_mem {x c : α} {ε : ℝ} (h : x ∈ ball c ε) : closedBall c ε ∈ 𝓝 x :=
mem_of_superset (isOpen_ball.mem_nhds h) ball_subset_closedBall
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
closedBall_mem_nhds_of_mem
| null |
nhdsWithin_basis_ball {s : Set α} :
(𝓝[s] x).HasBasis (fun ε : ℝ => 0 < ε) fun ε => ball x ε ∩ s :=
nhdsWithin_hasBasis nhds_basis_ball s
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
nhdsWithin_basis_ball
| null |
mem_nhdsWithin_iff {t : Set α} : s ∈ 𝓝[t] x ↔ ∃ ε > 0, ball x ε ∩ t ⊆ s :=
nhdsWithin_basis_ball.mem_iff
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
mem_nhdsWithin_iff
| null |
tendsto_nhdsWithin_nhdsWithin [PseudoMetricSpace β] {t : Set β} {f : α → β} {a b} :
Tendsto f (𝓝[s] a) (𝓝[t] b) ↔
∀ ε > 0, ∃ δ > 0, ∀ ⦃x : α⦄, x ∈ s → dist x a < δ → f x ∈ t ∧ dist (f x) b < ε :=
(nhdsWithin_basis_ball.tendsto_iff nhdsWithin_basis_ball).trans <| by
simp only [inter_comm _ s, inter_comm _ t, mem_inter_iff, and_imp, gt_iff_lt, mem_ball]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
tendsto_nhdsWithin_nhdsWithin
| null |
tendsto_nhdsWithin_nhds [PseudoMetricSpace β] {f : α → β} {a b} :
Tendsto f (𝓝[s] a) (𝓝 b) ↔
∀ ε > 0, ∃ δ > 0, ∀ ⦃x : α⦄, x ∈ s → dist x a < δ → dist (f x) b < ε := by
rw [← nhdsWithin_univ b, tendsto_nhdsWithin_nhdsWithin]
simp only [mem_univ, true_and]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
tendsto_nhdsWithin_nhds
| null |
tendsto_nhds_nhds [PseudoMetricSpace β] {f : α → β} {a b} :
Tendsto f (𝓝 a) (𝓝 b) ↔ ∀ ε > 0, ∃ δ > 0, ∀ ⦃x : α⦄, dist x a < δ → dist (f x) b < ε :=
nhds_basis_ball.tendsto_iff nhds_basis_ball
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
tendsto_nhds_nhds
| null |
continuousAt_iff [PseudoMetricSpace β] {f : α → β} {a : α} :
ContinuousAt f a ↔ ∀ ε > 0, ∃ δ > 0, ∀ ⦃x : α⦄, dist x a < δ → dist (f x) (f a) < ε := by
rw [ContinuousAt, tendsto_nhds_nhds]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
continuousAt_iff
| null |
continuousWithinAt_iff [PseudoMetricSpace β] {f : α → β} {a : α} {s : Set α} :
ContinuousWithinAt f s a ↔
∀ ε > 0, ∃ δ > 0, ∀ ⦃x : α⦄, x ∈ s → dist x a < δ → dist (f x) (f a) < ε := by
rw [ContinuousWithinAt, tendsto_nhdsWithin_nhds]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
continuousWithinAt_iff
| null |
continuousOn_iff [PseudoMetricSpace β] {f : α → β} {s : Set α} :
ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∃ δ > 0, ∀ a ∈ s, dist a b < δ → dist (f a) (f b) < ε := by
simp [ContinuousOn, continuousWithinAt_iff]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
continuousOn_iff
| null |
continuous_iff [PseudoMetricSpace β] {f : α → β} :
Continuous f ↔ ∀ b, ∀ ε > 0, ∃ δ > 0, ∀ a, dist a b < δ → dist (f a) (f b) < ε :=
continuous_iff_continuousAt.trans <| forall_congr' fun _ => tendsto_nhds_nhds
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
continuous_iff
| null |
tendsto_nhds {f : Filter β} {u : β → α} {a : α} :
Tendsto u f (𝓝 a) ↔ ∀ ε > 0, ∀ᶠ x in f, dist (u x) a < ε :=
nhds_basis_ball.tendsto_right_iff
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
tendsto_nhds
| null |
continuousAt_iff' [TopologicalSpace β] {f : β → α} {b : β} :
ContinuousAt f b ↔ ∀ ε > 0, ∀ᶠ x in 𝓝 b, dist (f x) (f b) < ε := by
rw [ContinuousAt, tendsto_nhds]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
continuousAt_iff'
| null |
continuousWithinAt_iff' [TopologicalSpace β] {f : β → α} {b : β} {s : Set β} :
ContinuousWithinAt f s b ↔ ∀ ε > 0, ∀ᶠ x in 𝓝[s] b, dist (f x) (f b) < ε := by
rw [ContinuousWithinAt, tendsto_nhds]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
continuousWithinAt_iff'
| null |
continuousOn_iff' [TopologicalSpace β] {f : β → α} {s : Set β} :
ContinuousOn f s ↔ ∀ b ∈ s, ∀ ε > 0, ∀ᶠ x in 𝓝[s] b, dist (f x) (f b) < ε := by
simp [ContinuousOn, continuousWithinAt_iff']
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
continuousOn_iff'
| null |
continuous_iff' [TopologicalSpace β] {f : β → α} :
Continuous f ↔ ∀ (a), ∀ ε > 0, ∀ᶠ x in 𝓝 a, dist (f x) (f a) < ε :=
continuous_iff_continuousAt.trans <| forall_congr' fun _ => tendsto_nhds
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
continuous_iff'
| null |
tendsto_atTop [Nonempty β] [SemilatticeSup β] {u : β → α} {a : α} :
Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n ≥ N, dist (u n) a < ε :=
(atTop_basis.tendsto_iff nhds_basis_ball).trans <| by
simp only [true_and, mem_ball, mem_Ici]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
tendsto_atTop
| null |
tendsto_atTop' [Nonempty β] [SemilatticeSup β] [NoMaxOrder β] {u : β → α} {a : α} :
Tendsto u atTop (𝓝 a) ↔ ∀ ε > 0, ∃ N, ∀ n > N, dist (u n) a < ε :=
(atTop_basis_Ioi.tendsto_iff nhds_basis_ball).trans <| by
simp only [true_and, gt_iff_lt, mem_Ioi, mem_ball]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
tendsto_atTop'
|
A variant of `tendsto_atTop` that
uses `∃ N, ∀ n > N, ...` rather than `∃ N, ∀ n ≥ N, ...`
|
isOpen_singleton_iff {α : Type*} [PseudoMetricSpace α] {x : α} :
IsOpen ({x} : Set α) ↔ ∃ ε > 0, ∀ y, dist y x < ε → y = x := by
simp [isOpen_iff, subset_singleton_iff, mem_ball]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
isOpen_singleton_iff
| null |
_root_.Dense.exists_dist_lt {s : Set α} (hs : Dense s) (x : α) {ε : ℝ} (hε : 0 < ε) :
∃ y ∈ s, dist x y < ε := by
have : (ball x ε).Nonempty := by simp [hε]
simpa only [mem_ball'] using hs.exists_mem_open isOpen_ball this
nonrec theorem _root_.DenseRange.exists_dist_lt {β : Type*} {f : β → α} (hf : DenseRange f) (x : α)
{ε : ℝ} (hε : 0 < ε) : ∃ y, dist x (f y) < ε :=
exists_range_iff.1 (hf.exists_dist_lt x hε)
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
_root_.Dense.exists_dist_lt
| null |
protected uniformSpace_eq_bot :
‹PseudoMetricSpace α›.toUniformSpace = ⊥ ↔
∃ r : ℝ, 0 < r ∧ Pairwise (r ≤ dist · · : α → α → Prop) := by
simp only [uniformity_basis_dist.uniformSpace_eq_bot, mem_setOf_eq, not_lt]
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
uniformSpace_eq_bot
|
(Pseudo) metric space has discrete `UniformSpace` structure
iff the distances between distinct points are uniformly bounded away from zero.
|
DiscreteTopology.of_forall_le_dist {α} [PseudoMetricSpace α] {r : ℝ} (hpos : 0 < r)
(hr : Pairwise (r ≤ dist · · : α → α → Prop)) : DiscreteTopology α :=
⟨by rw [Metric.uniformSpace_eq_bot.2 ⟨r, hpos, hr⟩, UniformSpace.toTopologicalSpace_bot]⟩
/- Instantiate a pseudometric space as a pseudoemetric space. Before we can state the instance,
we need to show that the uniform structure coming from the edistance and the
distance coincide. -/
|
lemma
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
DiscreteTopology.of_forall_le_dist
|
If the distances between distinct points in a (pseudo) metric space
are uniformly bounded away from zero, then the space has discrete topology.
|
Metric.uniformity_edist_aux {α} (d : α → α → ℝ≥0) :
⨅ ε > (0 : ℝ), 𝓟 { p : α × α | ↑(d p.1 p.2) < ε } =
⨅ ε > (0 : ℝ≥0∞), 𝓟 { p : α × α | ↑(d p.1 p.2) < ε } := by
simp only [le_antisymm_iff, le_iInf_iff, le_principal_iff]
refine ⟨fun ε hε => ?_, fun ε hε => ?_⟩
· rcases ENNReal.lt_iff_exists_nnreal_btwn.1 hε with ⟨ε', ε'0, ε'ε⟩
refine mem_iInf_of_mem (ε' : ℝ) (mem_iInf_of_mem (ENNReal.coe_pos.1 ε'0) ?_)
exact fun x hx => lt_trans (ENNReal.coe_lt_coe.2 hx) ε'ε
· lift ε to ℝ≥0 using le_of_lt hε
refine mem_iInf_of_mem (ε : ℝ≥0∞) (mem_iInf_of_mem (ENNReal.coe_pos.2 hε) ?_)
exact fun _ => ENNReal.coe_lt_coe.1
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Metric.uniformity_edist_aux
| null |
Metric.uniformity_edist : 𝓤 α = ⨅ ε > 0, 𝓟 { p : α × α | edist p.1 p.2 < ε } := by
simp only [PseudoMetricSpace.uniformity_dist, dist_nndist, edist_nndist,
Metric.uniformity_edist_aux]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Metric.uniformity_edist
| null |
Metric.eball_top_eq_univ (x : α) : EMetric.ball x ∞ = Set.univ :=
Set.eq_univ_iff_forall.mpr fun y => edist_lt_top y x
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Metric.eball_top_eq_univ
|
A pseudometric space induces a pseudoemetric space -/
instance (priority := 100) PseudoMetricSpace.toPseudoEMetricSpace : PseudoEMetricSpace α :=
{ ‹PseudoMetricSpace α› with
edist_self := by simp [edist_dist]
edist_comm := fun _ _ => by simp only [edist_dist, dist_comm]
edist_triangle := fun x y z => by
simp only [edist_dist, ← ENNReal.ofReal_add, dist_nonneg]
rw [ENNReal.ofReal_le_ofReal_iff _]
· exact dist_triangle _ _ _
· simpa using add_le_add (dist_nonneg : 0 ≤ dist x y) dist_nonneg
uniformity_edist := Metric.uniformity_edist }
/-- In a pseudometric space, an open ball of infinite radius is the whole space
|
@[simp]
Metric.emetric_ball {x : α} {ε : ℝ} : EMetric.ball x (ENNReal.ofReal ε) = ball x ε := by
ext y
simp only [EMetric.mem_ball, mem_ball, edist_dist]
exact ENNReal.ofReal_lt_ofReal_iff_of_nonneg dist_nonneg
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Metric.emetric_ball
|
Balls defined using the distance or the edistance coincide
|
@[simp]
Metric.emetric_ball_nnreal {x : α} {ε : ℝ≥0} : EMetric.ball x ε = ball x ε := by
rw [← Metric.emetric_ball]
simp
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Metric.emetric_ball_nnreal
|
Balls defined using the distance or the edistance coincide
|
Metric.emetric_closedBall {x : α} {ε : ℝ} (h : 0 ≤ ε) :
EMetric.closedBall x (ENNReal.ofReal ε) = closedBall x ε := by
ext y; simp [edist_le_ofReal h]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Metric.emetric_closedBall
|
Closed balls defined using the distance or the edistance coincide
|
@[simp]
Metric.emetric_closedBall_nnreal {x : α} {ε : ℝ≥0} :
EMetric.closedBall x ε = closedBall x ε := by
rw [← Metric.emetric_closedBall ε.coe_nonneg, ENNReal.ofReal_coe_nnreal]
@[simp]
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Metric.emetric_closedBall_nnreal
|
Closed balls defined using the distance or the edistance coincide
|
Metric.emetric_ball_top (x : α) : EMetric.ball x ⊤ = univ :=
eq_univ_of_forall fun _ => edist_lt_top _ _
|
theorem
|
Topology
|
[
"Mathlib.Data.ENNReal.Real",
"Mathlib.Tactic.Bound.Attribute",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.EMetricSpace.Defs",
"Mathlib.Topology.UniformSpace.Basic"
] |
Mathlib/Topology/MetricSpace/Pseudo/Defs.lean
|
Metric.emetric_ball_top
| null |
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