fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
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Filter.Tendsto.uniformity_trans {l : Filter β} {f₁ f₂ f₃ : β → α}
(h₁₂ : Tendsto (fun x => (f₁ x, f₂ x)) l (𝓤 α))
(h₂₃ : Tendsto (fun x => (f₂ x, f₃ x)) l (𝓤 α)) : Tendsto (fun x => (f₁ x, f₃ x)) l (𝓤 α) := by
refine le_trans (le_lift'.2 fun s hs => mem_map.2 ?_) comp_le_uniformity
filter_upwards [mem_map.1 (h₁₂ hs), mem_map.1 (h₂₃ hs)] with x hx₁₂ hx₂₃ using ⟨_, hx₁₂, hx₂₃⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | Filter.Tendsto.uniformity_trans | Relation `fun f g ↦ Tendsto (fun x ↦ (f x, g x)) l (𝓤 α)` is transitive. |
Filter.Tendsto.uniformity_symm {l : Filter β} {f : β → α × α} (h : Tendsto f l (𝓤 α)) :
Tendsto (fun x => ((f x).2, (f x).1)) l (𝓤 α) :=
tendsto_swap_uniformity.comp h | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | Filter.Tendsto.uniformity_symm | Relation `fun f g ↦ Tendsto (fun x ↦ (f x, g x)) l (𝓤 α)` is symmetric. |
tendsto_diag_uniformity (f : β → α) (l : Filter β) :
Tendsto (fun x => (f x, f x)) l (𝓤 α) := fun _s hs =>
mem_map.2 <| univ_mem' fun _ => refl_mem_uniformity hs | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | tendsto_diag_uniformity | Relation `fun f g ↦ Tendsto (fun x ↦ (f x, g x)) l (𝓤 α)` is reflexive. |
tendsto_const_uniformity {a : α} {f : Filter β} : Tendsto (fun _ => (a, a)) f (𝓤 α) :=
tendsto_diag_uniformity (fun _ => a) f | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | tendsto_const_uniformity | null |
symm_of_uniformity {s : Set (α × α)} (hs : s ∈ 𝓤 α) :
∃ t ∈ 𝓤 α, (∀ a b, (a, b) ∈ t → (b, a) ∈ t) ∧ t ⊆ s :=
have : preimage Prod.swap s ∈ 𝓤 α := symm_le_uniformity hs
⟨s ∩ preimage Prod.swap s, inter_mem hs this, fun _ _ ⟨h₁, h₂⟩ => ⟨h₂, h₁⟩, inter_subset_left⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | symm_of_uniformity | null |
comp_symm_of_uniformity {s : Set (α × α)} (hs : s ∈ 𝓤 α) :
∃ t ∈ 𝓤 α, (∀ {a b}, (a, b) ∈ t → (b, a) ∈ t) ∧ t ○ t ⊆ s :=
let ⟨_t, ht₁, ht₂⟩ := comp_mem_uniformity_sets hs
let ⟨t', ht', ht'₁, ht'₂⟩ := symm_of_uniformity ht₁
⟨t', ht', ht'₁ _ _, Subset.trans (monotone_id.compRel monotone_id ht'₂) ht₂⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | comp_symm_of_uniformity | null |
uniformity_le_symm : 𝓤 α ≤ map Prod.swap (𝓤 α) := by
rw [map_swap_eq_comap_swap]; exact tendsto_swap_uniformity.le_comap | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | uniformity_le_symm | null |
uniformity_eq_symm : 𝓤 α = map Prod.swap (𝓤 α) :=
le_antisymm uniformity_le_symm symm_le_uniformity
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | uniformity_eq_symm | null |
comap_swap_uniformity : comap (@Prod.swap α α) (𝓤 α) = 𝓤 α :=
(congr_arg _ uniformity_eq_symm).trans <| comap_map Prod.swap_injective | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | comap_swap_uniformity | null |
symmetrize_mem_uniformity {V : Set (α × α)} (h : V ∈ 𝓤 α) : symmetrizeRel V ∈ 𝓤 α := by
apply (𝓤 α).inter_sets h
rw [← image_swap_eq_preimage_swap, uniformity_eq_symm]
exact image_mem_map h | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | symmetrize_mem_uniformity | null |
UniformSpace.hasBasis_symmetric :
(𝓤 α).HasBasis (fun s : Set (α × α) => s ∈ 𝓤 α ∧ IsSymmetricRel s) id :=
hasBasis_self.2 fun t t_in =>
⟨symmetrizeRel t, symmetrize_mem_uniformity t_in, symmetric_symmetrizeRel t,
symmetrizeRel_subset_self t⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.hasBasis_symmetric | Symmetric entourages form a basis of `𝓤 α` |
uniformity_lift_le_swap {g : Set (α × α) → Filter β} {f : Filter β} (hg : Monotone g)
(h : ((𝓤 α).lift fun s => g (preimage Prod.swap s)) ≤ f) : (𝓤 α).lift g ≤ f :=
calc
(𝓤 α).lift g ≤ (Filter.map (@Prod.swap α α) <| 𝓤 α).lift g :=
lift_mono uniformity_le_symm le_rfl
_ ≤ _ := by rw [map_lift_eq2 hg, image_swap_eq_preimage_swap]; exact h | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | uniformity_lift_le_swap | null |
uniformity_lift_le_comp {f : Set (α × α) → Filter β} (h : Monotone f) :
((𝓤 α).lift fun s => f (s ○ s)) ≤ (𝓤 α).lift f :=
calc
((𝓤 α).lift fun s => f (s ○ s)) = ((𝓤 α).lift' fun s : Set (α × α) => s ○ s).lift f := by
rw [lift_lift'_assoc]
· exact monotone_id.compRel monotone_id
· exact h
_ ≤ (𝓤 α).lift f := lift_mono comp_le_uniformity le_rfl | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | uniformity_lift_le_comp | null |
comp3_mem_uniformity {s : Set (α × α)} (hs : s ∈ 𝓤 α) : ∃ t ∈ 𝓤 α, t ○ (t ○ t) ⊆ s :=
let ⟨_t', ht', ht's⟩ := comp_mem_uniformity_sets hs
let ⟨t, ht, htt'⟩ := comp_mem_uniformity_sets ht'
⟨t, ht, (compRel_mono ((subset_comp_self (refl_le_uniformity ht)).trans htt') htt').trans ht's⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | comp3_mem_uniformity | null |
comp_le_uniformity3 : ((𝓤 α).lift' fun s : Set (α × α) => s ○ (s ○ s)) ≤ 𝓤 α := fun _ h =>
let ⟨_t, htU, ht⟩ := comp3_mem_uniformity h
mem_of_superset (mem_lift' htU) ht | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | comp_le_uniformity3 | See also `comp3_mem_uniformity`. |
comp_symm_mem_uniformity_sets {s : Set (α × α)} (hs : s ∈ 𝓤 α) :
∃ t ∈ 𝓤 α, IsSymmetricRel t ∧ t ○ t ⊆ s := by
obtain ⟨w, w_in, w_sub⟩ : ∃ w ∈ 𝓤 α, w ○ w ⊆ s := comp_mem_uniformity_sets hs
use symmetrizeRel w, symmetrize_mem_uniformity w_in, symmetric_symmetrizeRel w
have : symmetrizeRel w ⊆ w := symmetrizeRel_subset_self w
calc symmetrizeRel w ○ symmetrizeRel w
_ ⊆ w ○ w := by gcongr
_ ⊆ s := w_sub | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | comp_symm_mem_uniformity_sets | See also `comp_open_symm_mem_uniformity_sets`. |
subset_comp_self_of_mem_uniformity {s : Set (α × α)} (h : s ∈ 𝓤 α) : s ⊆ s ○ s :=
subset_comp_self (refl_le_uniformity h) | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | subset_comp_self_of_mem_uniformity | null |
comp_comp_symm_mem_uniformity_sets {s : Set (α × α)} (hs : s ∈ 𝓤 α) :
∃ t ∈ 𝓤 α, IsSymmetricRel t ∧ t ○ t ○ t ⊆ s := by
rcases comp_symm_mem_uniformity_sets hs with ⟨w, w_in, _, w_sub⟩
rcases comp_symm_mem_uniformity_sets w_in with ⟨t, t_in, t_symm, t_sub⟩
use t, t_in, t_symm
have : t ⊆ t ○ t := subset_comp_self_of_mem_uniformity t_in
calc
t ○ t ○ t ⊆ w ○ (t ○ t) := by gcongr
_ ⊆ w ○ w := by gcongr
_ ⊆ s := w_sub
/-! | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | comp_comp_symm_mem_uniformity_sets | null |
ball (x : β) (V : Set (β × β)) : Set β := Prod.mk x ⁻¹' V
open UniformSpace (ball) | def | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | ball | The ball around `(x : β)` with respect to `(V : Set (β × β))`. Intended to be
used for `V ∈ 𝓤 β`, but this is not needed for the definition. Recovers the
notions of metric space ball when `V = {p | dist p.1 p.2 < r }`. |
mem_ball_self (x : α) {V : Set (α × α)} : V ∈ 𝓤 α → x ∈ ball x V := refl_mem_uniformity | lemma | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | mem_ball_self | null |
mem_ball_comp {V W : Set (β × β)} {x y z} (h : y ∈ ball x V) (h' : z ∈ ball y W) :
z ∈ ball x (V ○ W) :=
prodMk_mem_compRel h h' | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | mem_ball_comp | The triangle inequality for `UniformSpace.ball` |
ball_subset_of_comp_subset {V W : Set (β × β)} {x y} (h : x ∈ ball y W) (h' : W ○ W ⊆ V) :
ball x W ⊆ ball y V := fun _z z_in => h' (mem_ball_comp h z_in) | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | ball_subset_of_comp_subset | null |
ball_mono {V W : Set (β × β)} (h : V ⊆ W) (x : β) : ball x V ⊆ ball x W :=
preimage_mono h | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | ball_mono | null |
ball_inter (x : β) (V W : Set (β × β)) : ball x (V ∩ W) = ball x V ∩ ball x W :=
preimage_inter | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | ball_inter | null |
ball_inter_left (x : β) (V W : Set (β × β)) : ball x (V ∩ W) ⊆ ball x V :=
ball_mono inter_subset_left x | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | ball_inter_left | null |
ball_inter_right (x : β) (V W : Set (β × β)) : ball x (V ∩ W) ⊆ ball x W :=
ball_mono inter_subset_right x | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | ball_inter_right | null |
ball_iInter {x : β} {V : ι → Set (β × β)} : ball x (⋂ i, V i) = ⋂ i, ball x (V i) :=
preimage_iInter | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | ball_iInter | null |
mem_ball_symmetry {V : Set (β × β)} (hV : IsSymmetricRel V) {x y} :
x ∈ ball y V ↔ y ∈ ball x V :=
show (x, y) ∈ Prod.swap ⁻¹' V ↔ (x, y) ∈ V by
unfold IsSymmetricRel at hV
rw [hV] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | mem_ball_symmetry | null |
ball_eq_of_symmetry {V : Set (β × β)} (hV : IsSymmetricRel V) {x} :
ball x V = { y | (y, x) ∈ V } := by
ext y
rw [mem_ball_symmetry hV]
exact Iff.rfl | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | ball_eq_of_symmetry | null |
mem_comp_of_mem_ball {V W : Set (β × β)} {x y z : β} (hV : IsSymmetricRel V)
(hx : x ∈ ball z V) (hy : y ∈ ball z W) : (x, y) ∈ V ○ W := by
rw [mem_ball_symmetry hV] at hx
exact ⟨z, hx, hy⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | mem_comp_of_mem_ball | null |
mem_comp_comp {V W M : Set (β × β)} (hW' : IsSymmetricRel W) {p : β × β} :
p ∈ V ○ M ○ W ↔ (ball p.1 V ×ˢ ball p.2 W ∩ M).Nonempty := by
obtain ⟨x, y⟩ := p
constructor
· rintro ⟨z, ⟨w, hpw, hwz⟩, hzy⟩
exact ⟨(w, z), ⟨hpw, by rwa [mem_ball_symmetry hW']⟩, hwz⟩
· rintro ⟨⟨w, z⟩, ⟨w_in, z_in⟩, hwz⟩
rw [mem_ball_symmetry hW'] at z_in
exact ⟨z, ⟨w, w_in, hwz⟩, z_in⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | mem_comp_comp | null |
mem_nhds_uniformity_iff_right {x : α} {s : Set α} :
s ∈ 𝓝 x ↔ { p : α × α | p.1 = x → p.2 ∈ s } ∈ 𝓤 α := by
simp only [nhds_eq_comap_uniformity, mem_comap_prodMk] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | mem_nhds_uniformity_iff_right | null |
mem_nhds_uniformity_iff_left {x : α} {s : Set α} :
s ∈ 𝓝 x ↔ { p : α × α | p.2 = x → p.1 ∈ s } ∈ 𝓤 α := by
rw [uniformity_eq_symm, mem_nhds_uniformity_iff_right]
simp only [mem_map, preimage_setOf_eq, Prod.snd_swap, Prod.fst_swap] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | mem_nhds_uniformity_iff_left | null |
nhdsWithin_eq_comap_uniformity_of_mem {x : α} {T : Set α} (hx : x ∈ T) (S : Set α) :
𝓝[S] x = (𝓤 α ⊓ 𝓟 (T ×ˢ S)).comap (Prod.mk x) := by
simp [nhdsWithin, nhds_eq_comap_uniformity, hx] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | nhdsWithin_eq_comap_uniformity_of_mem | null |
nhdsWithin_eq_comap_uniformity {x : α} (S : Set α) :
𝓝[S] x = (𝓤 α ⊓ 𝓟 (univ ×ˢ S)).comap (Prod.mk x) :=
nhdsWithin_eq_comap_uniformity_of_mem (mem_univ _) S | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | nhdsWithin_eq_comap_uniformity | null |
isOpen_iff_ball_subset {s : Set α} : IsOpen s ↔ ∀ x ∈ s, ∃ V ∈ 𝓤 α, ball x V ⊆ s := by
simp_rw [isOpen_iff_mem_nhds, nhds_eq_comap_uniformity, mem_comap, ball] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | isOpen_iff_ball_subset | See also `isOpen_iff_isOpen_ball_subset`. |
nhds_basis_uniformity' {p : ι → Prop} {s : ι → Set (α × α)} (h : (𝓤 α).HasBasis p s)
{x : α} : (𝓝 x).HasBasis p fun i => ball x (s i) := by
rw [nhds_eq_comap_uniformity]
exact h.comap (Prod.mk x) | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | nhds_basis_uniformity' | null |
nhds_basis_uniformity {p : ι → Prop} {s : ι → Set (α × α)} (h : (𝓤 α).HasBasis p s)
{x : α} : (𝓝 x).HasBasis p fun i => { y | (y, x) ∈ s i } := by
replace h := h.comap Prod.swap
rw [comap_swap_uniformity] at h
exact nhds_basis_uniformity' h | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | nhds_basis_uniformity | null |
nhds_eq_comap_uniformity' {x : α} : 𝓝 x = (𝓤 α).comap fun y => (y, x) :=
(nhds_basis_uniformity (𝓤 α).basis_sets).eq_of_same_basis <| (𝓤 α).basis_sets.comap _ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | nhds_eq_comap_uniformity' | null |
UniformSpace.mem_nhds_iff {x : α} {s : Set α} : s ∈ 𝓝 x ↔ ∃ V ∈ 𝓤 α, ball x V ⊆ s := by
rw [nhds_eq_comap_uniformity, mem_comap]
simp_rw [ball] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.mem_nhds_iff | null |
UniformSpace.ball_mem_nhds (x : α) ⦃V : Set (α × α)⦄ (V_in : V ∈ 𝓤 α) : ball x V ∈ 𝓝 x := by
rw [UniformSpace.mem_nhds_iff]
exact ⟨V, V_in, Subset.rfl⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.ball_mem_nhds | null |
UniformSpace.ball_mem_nhdsWithin {x : α} {S : Set α} ⦃V : Set (α × α)⦄ (x_in : x ∈ S)
(V_in : V ∈ 𝓤 α ⊓ 𝓟 (S ×ˢ S)) : ball x V ∈ 𝓝[S] x := by
rw [nhdsWithin_eq_comap_uniformity_of_mem x_in, mem_comap]
exact ⟨V, V_in, Subset.rfl⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.ball_mem_nhdsWithin | null |
UniformSpace.mem_nhds_iff_symm {x : α} {s : Set α} :
s ∈ 𝓝 x ↔ ∃ V ∈ 𝓤 α, IsSymmetricRel V ∧ ball x V ⊆ s := by
rw [UniformSpace.mem_nhds_iff]
constructor
· rintro ⟨V, V_in, V_sub⟩
use symmetrizeRel V, symmetrize_mem_uniformity V_in, symmetric_symmetrizeRel V
exact Subset.trans (ball_mono (symmetrizeRel_subset_self V) x) V_sub
· rintro ⟨V, V_in, _, V_sub⟩
exact ⟨V, V_in, V_sub⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.mem_nhds_iff_symm | null |
UniformSpace.hasBasis_nhds (x : α) :
HasBasis (𝓝 x) (fun s : Set (α × α) => s ∈ 𝓤 α ∧ IsSymmetricRel s) fun s => ball x s :=
⟨fun t => by simp [UniformSpace.mem_nhds_iff_symm, and_assoc]⟩
open UniformSpace | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.hasBasis_nhds | null |
UniformSpace.mem_closure_iff_symm_ball {s : Set α} {x} :
x ∈ closure s ↔ ∀ {V}, V ∈ 𝓤 α → IsSymmetricRel V → (s ∩ ball x V).Nonempty := by
simp [mem_closure_iff_nhds_basis (hasBasis_nhds x), Set.Nonempty] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.mem_closure_iff_symm_ball | null |
UniformSpace.mem_closure_iff_ball {s : Set α} {x} :
x ∈ closure s ↔ ∀ {V}, V ∈ 𝓤 α → (ball x V ∩ s).Nonempty := by
simp [mem_closure_iff_nhds_basis' (nhds_basis_uniformity' (𝓤 α).basis_sets)] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.mem_closure_iff_ball | null |
nhds_eq_uniformity {x : α} : 𝓝 x = (𝓤 α).lift' (ball x) :=
(nhds_basis_uniformity' (𝓤 α).basis_sets).eq_biInf | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | nhds_eq_uniformity | null |
nhds_eq_uniformity' {x : α} : 𝓝 x = (𝓤 α).lift' fun s => { y | (y, x) ∈ s } :=
(nhds_basis_uniformity (𝓤 α).basis_sets).eq_biInf | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | nhds_eq_uniformity' | null |
mem_nhds_left (x : α) {s : Set (α × α)} (h : s ∈ 𝓤 α) : { y : α | (x, y) ∈ s } ∈ 𝓝 x :=
ball_mem_nhds x h | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | mem_nhds_left | null |
mem_nhds_right (y : α) {s : Set (α × α)} (h : s ∈ 𝓤 α) : { x : α | (x, y) ∈ s } ∈ 𝓝 y :=
mem_nhds_left _ (symm_le_uniformity h) | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | mem_nhds_right | null |
exists_mem_nhds_ball_subset_of_mem_nhds {a : α} {U : Set α} (h : U ∈ 𝓝 a) :
∃ V ∈ 𝓝 a, ∃ t ∈ 𝓤 α, ∀ a' ∈ V, UniformSpace.ball a' t ⊆ U :=
let ⟨t, ht, htU⟩ := comp_mem_uniformity_sets (mem_nhds_uniformity_iff_right.1 h)
⟨_, mem_nhds_left a ht, t, ht, fun a₁ h₁ a₂ h₂ => @htU (a, a₂) ⟨a₁, h₁, h₂⟩ rfl⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | exists_mem_nhds_ball_subset_of_mem_nhds | null |
tendsto_right_nhds_uniformity {a : α} : Tendsto (fun a' => (a', a)) (𝓝 a) (𝓤 α) := fun _ =>
mem_nhds_right a | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | tendsto_right_nhds_uniformity | null |
tendsto_left_nhds_uniformity {a : α} : Tendsto (fun a' => (a, a')) (𝓝 a) (𝓤 α) := fun _ =>
mem_nhds_left a | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | tendsto_left_nhds_uniformity | null |
lift_nhds_left {x : α} {g : Set α → Filter β} (hg : Monotone g) :
(𝓝 x).lift g = (𝓤 α).lift fun s : Set (α × α) => g (ball x s) := by
rw [nhds_eq_comap_uniformity, comap_lift_eq2 hg]
simp_rw [ball, Function.comp_def] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | lift_nhds_left | null |
lift_nhds_right {x : α} {g : Set α → Filter β} (hg : Monotone g) :
(𝓝 x).lift g = (𝓤 α).lift fun s : Set (α × α) => g { y | (y, x) ∈ s } := by
rw [nhds_eq_comap_uniformity', comap_lift_eq2 hg]
simp_rw [Function.comp_def, preimage] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | lift_nhds_right | null |
nhds_nhds_eq_uniformity_uniformity_prod {a b : α} :
𝓝 a ×ˢ 𝓝 b = (𝓤 α).lift fun s : Set (α × α) =>
(𝓤 α).lift' fun t => { y : α | (y, a) ∈ s } ×ˢ { y : α | (b, y) ∈ t } := by
rw [nhds_eq_uniformity', nhds_eq_uniformity, prod_lift'_lift']
exacts [rfl, monotone_preimage, monotone_preimage] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | nhds_nhds_eq_uniformity_uniformity_prod | null |
Filter.HasBasis.biInter_biUnion_ball {p : ι → Prop} {U : ι → Set (α × α)}
(h : HasBasis (𝓤 α) p U) (s : Set α) :
(⋂ (i) (_ : p i), ⋃ x ∈ s, ball x (U i)) = closure s := by
ext x
simp [mem_closure_iff_nhds_basis (nhds_basis_uniformity h), ball]
/-! ### Uniform continuity -/ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | Filter.HasBasis.biInter_biUnion_ball | null |
UniformContinuous [UniformSpace β] (f : α → β) :=
Tendsto (fun x : α × α => (f x.1, f x.2)) (𝓤 α) (𝓤 β) | def | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformContinuous | A function `f : α → β` is *uniformly continuous* if `(f x, f y)` tends to the diagonal
as `(x, y)` tends to the diagonal. In other words, if `x` is sufficiently close to `y`, then
`f x` is close to `f y` no matter where `x` and `y` are located in `α`. |
UniformContinuousOn [UniformSpace β] (f : α → β) (s : Set α) : Prop :=
Tendsto (fun x : α × α => (f x.1, f x.2)) (𝓤 α ⊓ 𝓟 (s ×ˢ s)) (𝓤 β) | def | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformContinuousOn | Notation for uniform continuity with respect to non-standard `UniformSpace` instances. -/
scoped[Uniformity] notation "UniformContinuous[" u₁ ", " u₂ "]" => @UniformContinuous _ _ u₁ u₂
/-- A function `f : α → β` is *uniformly continuous* on `s : Set α` if `(f x, f y)` tends to
the diagonal as `(x, y)` tends to the diagonal while remaining in `s ×ˢ s`.
In other words, if `x` is sufficiently close to `y`, then `f x` is close to
`f y` no matter where `x` and `y` are located in `s`. |
uniformContinuous_def [UniformSpace β] {f : α → β} :
UniformContinuous f ↔ ∀ r ∈ 𝓤 β, { x : α × α | (f x.1, f x.2) ∈ r } ∈ 𝓤 α :=
Iff.rfl | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | uniformContinuous_def | null |
uniformContinuous_iff_eventually [UniformSpace β] {f : α → β} :
UniformContinuous f ↔ ∀ r ∈ 𝓤 β, ∀ᶠ x : α × α in 𝓤 α, (f x.1, f x.2) ∈ r :=
Iff.rfl | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | uniformContinuous_iff_eventually | null |
uniformContinuousOn_univ [UniformSpace β] {f : α → β} :
UniformContinuousOn f univ ↔ UniformContinuous f := by
rw [UniformContinuousOn, UniformContinuous, univ_prod_univ, principal_univ, inf_top_eq] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | uniformContinuousOn_univ | null |
uniformContinuous_of_const [UniformSpace β] {c : α → β} (h : ∀ a b, c a = c b) :
UniformContinuous c :=
have : (fun x : α × α => (c x.fst, c x.snd)) ⁻¹' idRel = univ :=
eq_univ_iff_forall.2 fun ⟨a, b⟩ => h a b
le_trans (map_le_iff_le_comap.2 <| by simp [comap_principal, this]) refl_le_uniformity | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | uniformContinuous_of_const | null |
uniformContinuous_id : UniformContinuous (@id α) := tendsto_id | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | uniformContinuous_id | null |
uniformContinuous_const [UniformSpace β] {b : β} : UniformContinuous fun _ : α => b :=
uniformContinuous_of_const fun _ _ => rfl
nonrec theorem UniformContinuous.comp [UniformSpace β] [UniformSpace γ] {g : β → γ} {f : α → β}
(hg : UniformContinuous g) (hf : UniformContinuous f) : UniformContinuous (g ∘ f) :=
hg.comp hf | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | uniformContinuous_const | null |
UniformContinuous.iterate [UniformSpace β] (T : β → β) (n : ℕ) (h : UniformContinuous T) :
UniformContinuous T^[n] := by
induction n with
| zero => exact uniformContinuous_id
| succ n hn => exact Function.iterate_succ _ _ ▸ UniformContinuous.comp hn h | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformContinuous.iterate | If a function `T` is uniformly continuous in a uniform space `β`,
then its `n`-th iterate `T^[n]` is also uniformly continuous. |
Filter.HasBasis.uniformContinuous_iff {ι'} [UniformSpace β] {p : ι → Prop}
{s : ι → Set (α × α)} (ha : (𝓤 α).HasBasis p s) {q : ι' → Prop} {t : ι' → Set (β × β)}
(hb : (𝓤 β).HasBasis q t) {f : α → β} :
UniformContinuous f ↔ ∀ i, q i → ∃ j, p j ∧ ∀ x y, (x, y) ∈ s j → (f x, f y) ∈ t i :=
(ha.tendsto_iff hb).trans <| by simp only [Prod.forall] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | Filter.HasBasis.uniformContinuous_iff | null |
Filter.HasBasis.uniformContinuousOn_iff {ι'} [UniformSpace β] {p : ι → Prop}
{s : ι → Set (α × α)} (ha : (𝓤 α).HasBasis p s) {q : ι' → Prop} {t : ι' → Set (β × β)}
(hb : (𝓤 β).HasBasis q t) {f : α → β} {S : Set α} :
UniformContinuousOn f S ↔
∀ i, q i → ∃ j, p j ∧ ∀ x, x ∈ S → ∀ y, y ∈ S → (x, y) ∈ s j → (f x, f y) ∈ t i :=
((ha.inf_principal (S ×ˢ S)).tendsto_iff hb).trans <| by
simp_rw [Prod.forall, Set.inter_comm (s _), forall_mem_comm, mem_inter_iff, mem_prod, and_imp] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | Filter.HasBasis.uniformContinuousOn_iff | null |
tendstoLocallyUniformly_of_forall_tendsto
(hF_cont : ∀ i, Continuous (F i)) (hF_mono : Monotone F) (hf : Continuous f)
(h_tendsto : ∀ x, Tendsto (F · x) atTop (𝓝 (f x))) :
TendstoLocallyUniformly F f atTop := by
refine (atTop : Filter ι).eq_or_neBot.elim (fun h ↦ ?eq_bot) (fun _ ↦ ?_)
case eq_bot => simp [h, tendstoLocallyUniformly_iff_forall_tendsto]
have F_le_f (x : α) (n : ι) : F n x ≤ f x := by
refine ge_of_tendsto (h_tendsto x) ?_
filter_upwards [Ici_mem_atTop n] with m hnm
exact hF_mono hnm x
simp_rw [Metric.tendstoLocallyUniformly_iff, dist_eq_norm']
intro ε ε_pos x
simp_rw +singlePass [tendsto_iff_norm_sub_tendsto_zero] at h_tendsto
obtain ⟨n, hn⟩ := (h_tendsto x).eventually (eventually_lt_nhds ε_pos) |>.exists
refine ⟨{y | ‖F n y - f y‖ < ε}, ⟨isOpen_lt (by fun_prop) continuous_const |>.mem_nhds hn, ?_⟩⟩
filter_upwards [eventually_ge_atTop n] with m hnm z hz
refine norm_le_norm_of_abs_le_abs ?_ |>.trans_lt hz
simp only [abs_of_nonpos (sub_nonpos_of_le (F_le_f _ _)), neg_sub, sub_le_sub_iff_left]
exact hF_mono hnm z | lemma | Topology | [
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/UniformSpace/Dini.lean | tendstoLocallyUniformly_of_forall_tendsto | **Dini's theorem**: if `F n` is a monotone increasing collection of continuous functions
converging pointwise to a continuous function `f`, then `F n` converges locally uniformly to `f`. |
tendstoLocallyUniformlyOn_of_forall_tendsto {s : Set α}
(hF_cont : ∀ i, ContinuousOn (F i) s) (hF_mono : ∀ x ∈ s, Monotone (F · x))
(hf : ContinuousOn f s) (h_tendsto : ∀ x ∈ s, Tendsto (F · x) atTop (𝓝 (f x))) :
TendstoLocallyUniformlyOn F f atTop s := by
rw [tendstoLocallyUniformlyOn_iff_tendstoLocallyUniformly_comp_coe]
exact tendstoLocallyUniformly_of_forall_tendsto (hF_cont · |>.restrict)
(fun _ _ h x ↦ hF_mono _ x.2 h) hf.restrict (fun x ↦ h_tendsto x x.2) | lemma | Topology | [
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/UniformSpace/Dini.lean | tendstoLocallyUniformlyOn_of_forall_tendsto | **Dini's theorem**: if `F n` is a monotone increasing collection of continuous functions on a
set `s` converging pointwise to a continuous function `f`, then `F n` converges locally uniformly
to `f`. |
tendstoUniformly_of_forall_tendsto [CompactSpace α] (hF_cont : ∀ i, Continuous (F i))
(hF_mono : Monotone F) (hf : Continuous f) (h_tendsto : ∀ x, Tendsto (F · x) atTop (𝓝 (f x))) :
TendstoUniformly F f atTop :=
tendstoLocallyUniformly_iff_tendstoUniformly_of_compactSpace.mp <|
tendstoLocallyUniformly_of_forall_tendsto hF_cont hF_mono hf h_tendsto | lemma | Topology | [
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/UniformSpace/Dini.lean | tendstoUniformly_of_forall_tendsto | **Dini's theorem**: if `F n` is a monotone increasing collection of continuous functions on a
compact space converging pointwise to a continuous function `f`, then `F n` converges uniformly to
`f`. |
tendstoUniformlyOn_of_forall_tendsto {s : Set α} (hs : IsCompact s)
(hF_cont : ∀ i, ContinuousOn (F i) s) (hF_mono : ∀ x ∈ s, Monotone (F · x))
(hf : ContinuousOn f s) (h_tendsto : ∀ x ∈ s, Tendsto (F · x) atTop (𝓝 (f x))) :
TendstoUniformlyOn F f atTop s :=
tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact hs |>.mp <|
tendstoLocallyUniformlyOn_of_forall_tendsto hF_cont hF_mono hf h_tendsto | lemma | Topology | [
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/UniformSpace/Dini.lean | tendstoUniformlyOn_of_forall_tendsto | **Dini's theorem**: if `F n` is a monotone increasing collection of continuous functions on a
compact set `s` converging pointwise to a continuous function `f`, then `F n` converges uniformly
to `f`. |
tendstoLocallyUniformly_of_forall_tendsto
(hF_cont : ∀ i, Continuous (F i)) (hF_anti : Antitone F) (hf : Continuous f)
(h_tendsto : ∀ x, Tendsto (F · x) atTop (𝓝 (f x))) :
TendstoLocallyUniformly F f atTop :=
Monotone.tendstoLocallyUniformly_of_forall_tendsto (G := Gᵒᵈ) hF_cont hF_anti hf h_tendsto | lemma | Topology | [
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/UniformSpace/Dini.lean | tendstoLocallyUniformly_of_forall_tendsto | **Dini's theorem**: if `F n` is a monotone decreasing collection of continuous functions on a
converging pointwise to a continuous function `f`, then `F n` converges locally uniformly to `f`. |
tendstoLocallyUniformlyOn_of_forall_tendsto {s : Set α}
(hF_cont : ∀ i, ContinuousOn (F i) s) (hF_anti : ∀ x ∈ s, Antitone (F · x))
(hf : ContinuousOn f s) (h_tendsto : ∀ x ∈ s, Tendsto (F · x) atTop (𝓝 (f x))) :
TendstoLocallyUniformlyOn F f atTop s :=
Monotone.tendstoLocallyUniformlyOn_of_forall_tendsto (G := Gᵒᵈ) hF_cont hF_anti hf h_tendsto | lemma | Topology | [
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/UniformSpace/Dini.lean | tendstoLocallyUniformlyOn_of_forall_tendsto | **Dini's theorem**: if `F n` is a monotone decreasing collection of continuous functions on a
set `s` converging pointwise to a continuous function `f`, then `F n` converges locally uniformly
to `f`. |
tendstoUniformly_of_forall_tendsto [CompactSpace α] (hF_cont : ∀ i, Continuous (F i))
(hF_anti : Antitone F) (hf : Continuous f) (h_tendsto : ∀ x, Tendsto (F · x) atTop (𝓝 (f x))) :
TendstoUniformly F f atTop :=
Monotone.tendstoUniformly_of_forall_tendsto (G := Gᵒᵈ) hF_cont hF_anti hf h_tendsto | lemma | Topology | [
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/UniformSpace/Dini.lean | tendstoUniformly_of_forall_tendsto | **Dini's theorem**: if `F n` is a monotone decreasing collection of continuous functions on a
compact space converging pointwise to a continuous function `f`, then `F n` converges uniformly
to `f`. |
tendstoUniformlyOn_of_forall_tendsto {s : Set α} (hs : IsCompact s)
(hF_cont : ∀ i, ContinuousOn (F i) s) (hF_anti : ∀ x ∈ s, Antitone (F · x))
(hf : ContinuousOn f s) (h_tendsto : ∀ x ∈ s, Tendsto (F · x) atTop (𝓝 (f x))) :
TendstoUniformlyOn F f atTop s :=
Monotone.tendstoUniformlyOn_of_forall_tendsto (G := Gᵒᵈ) hs hF_cont hF_anti hf h_tendsto | lemma | Topology | [
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/UniformSpace/Dini.lean | tendstoUniformlyOn_of_forall_tendsto | **Dini's theorem**: if `F n` is a monotone decreasing collection of continuous functions on a
compact set `s` converging pointwise to a continuous `f`, then `F n` converges uniformly to `f`. |
tendsto_of_monotone_of_pointwise (hF_mono : Monotone F)
(h_tendsto : ∀ x, Tendsto (F · x) atTop (𝓝 (f x))) :
Tendsto F atTop (𝓝 f) :=
tendsto_of_tendstoLocallyUniformly <|
hF_mono.tendstoLocallyUniformly_of_forall_tendsto (F · |>.continuous) f.continuous h_tendsto | lemma | Topology | [
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/UniformSpace/Dini.lean | tendsto_of_monotone_of_pointwise | **Dini's theorem**: if `F n` is a monotone increasing collection of continuous functions
converging pointwise to a continuous function `f`, then `F n` converges to `f` in the
compact-open topology. |
tendsto_of_antitone_of_pointwise (hF_anti : Antitone F)
(h_tendsto : ∀ x, Tendsto (F · x) atTop (𝓝 (f x))) :
Tendsto F atTop (𝓝 f) :=
tendsto_of_monotone_of_pointwise (G := Gᵒᵈ) hF_anti h_tendsto | lemma | Topology | [
"Mathlib.Analysis.Normed.Order.Lattice",
"Mathlib.Topology.ContinuousMap.Ordered",
"Mathlib.Topology.UniformSpace.CompactConvergence"
] | Mathlib/Topology/UniformSpace/Dini.lean | tendsto_of_antitone_of_pointwise | **Dini's theorem**: if `F n` is a monotone decreasing collection of continuous functions
converging pointwise to a continuous function `f`, then `F n` converges to `f` in the
compact-open topology. |
@[mk_iff discreteUniformity_iff_eq_bot]
DiscreteUniformity (X : Type*) [u : UniformSpace X] : Prop where
eq_bot : u = ⊥ | class | Topology | [
"Mathlib.Topology.UniformSpace.Basic"
] | Mathlib/Topology/UniformSpace/DiscreteUniformity.lean | DiscreteUniformity | The discrete uniformity |
uniformContinuous {Y : Type*} [UniformSpace Y] (f : X → Y) :
UniformContinuous f := by
simp only [uniformContinuous_iff, DiscreteUniformity.eq_bot, bot_le] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.Basic"
] | Mathlib/Topology/UniformSpace/DiscreteUniformity.lean | uniformContinuous | The bot uniformity is the discrete uniformity. -/
instance (X : Type*) : @DiscreteUniformity X ⊥ :=
@DiscreteUniformity.mk X ⊥ rfl
variable (X : Type*) [u : UniformSpace X] [DiscreteUniformity X]
theorem _root_.discreteUniformity_iff_eq_principal_idRel {X : Type*} [UniformSpace X] :
DiscreteUniformity X ↔ uniformity X = principal idRel := by
rw [discreteUniformity_iff_eq_bot, UniformSpace.ext_iff, Filter.ext_iff, bot_uniformity]
theorem eq_principal_idRel : uniformity X = principal idRel :=
discreteUniformity_iff_eq_principal_idRel.mp inferInstance
/-- The discrete uniformity induces the discrete topology. -/
instance : DiscreteTopology X where
eq_bot := by
rw [DiscreteUniformity.eq_bot (X := X), UniformSpace.toTopologicalSpace_bot]
theorem _root_.discreteUniformity_iff_idRel_mem_uniformity {X : Type*} [UniformSpace X] :
DiscreteUniformity X ↔ idRel ∈ uniformity X := by
rw [← uniformSpace_eq_bot, discreteUniformity_iff_eq_bot]
theorem idRel_mem_uniformity : idRel ∈ uniformity X :=
discreteUniformity_iff_idRel_mem_uniformity.mp inferInstance
variable {X} in
/-- A product of spaces with discrete uniformity has a discrete uniformity. -/
instance {Y : Type*} [UniformSpace Y] [DiscreteUniformity Y] :
DiscreteUniformity (X × Y) := by
simp [discreteUniformity_iff_eq_principal_idRel, uniformity_prod_eq_comap_prod,
eq_principal_idRel, idRel, Set.prod_eq, Prod.ext_iff, Set.setOf_and]
variable {x} in
/-- On a space with a discrete uniformity, any function is uniformly continuous. |
EquicontinuousAt (F : ι → X → α) (x₀ : X) : Prop :=
∀ U ∈ 𝓤 α, ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ U | def | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | EquicontinuousAt | A family `F : ι → X → α` of functions from a topological space to a uniform space is
*equicontinuous at `x₀ : X`* if, for all entourages `U ∈ 𝓤 α`, there is a neighborhood `V` of `x₀`
such that, for all `x ∈ V` and for all `i : ι`, `F i x` is `U`-close to `F i x₀`. |
protected Set.EquicontinuousAt (H : Set <| X → α) (x₀ : X) : Prop :=
EquicontinuousAt ((↑) : H → X → α) x₀ | abbrev | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.EquicontinuousAt | We say that a set `H : Set (X → α)` of functions is equicontinuous at a point if the family
`(↑) : ↥H → (X → α)` is equicontinuous at that point. |
EquicontinuousWithinAt (F : ι → X → α) (S : Set X) (x₀ : X) : Prop :=
∀ U ∈ 𝓤 α, ∀ᶠ x in 𝓝[S] x₀, ∀ i, (F i x₀, F i x) ∈ U | def | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | EquicontinuousWithinAt | A family `F : ι → X → α` of functions from a topological space to a uniform space is
*equicontinuous at `x₀ : X` within `S : Set X`* if, for all entourages `U ∈ 𝓤 α`, there is a
neighborhood `V` of `x₀` within `S` such that, for all `x ∈ V` and for all `i : ι`, `F i x` is
`U`-close to `F i x₀`. |
protected Set.EquicontinuousWithinAt (H : Set <| X → α) (S : Set X) (x₀ : X) : Prop :=
EquicontinuousWithinAt ((↑) : H → X → α) S x₀ | abbrev | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.EquicontinuousWithinAt | We say that a set `H : Set (X → α)` of functions is equicontinuous at a point within a subset
if the family `(↑) : ↥H → (X → α)` is equicontinuous at that point within that same subset. |
Equicontinuous (F : ι → X → α) : Prop :=
∀ x₀, EquicontinuousAt F x₀ | def | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Equicontinuous | A family `F : ι → X → α` of functions from a topological space to a uniform space is
*equicontinuous* on all of `X` if it is equicontinuous at each point of `X`. |
protected Set.Equicontinuous (H : Set <| X → α) : Prop :=
Equicontinuous ((↑) : H → X → α) | abbrev | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.Equicontinuous | We say that a set `H : Set (X → α)` of functions is equicontinuous if the family
`(↑) : ↥H → (X → α)` is equicontinuous. |
EquicontinuousOn (F : ι → X → α) (S : Set X) : Prop :=
∀ x₀ ∈ S, EquicontinuousWithinAt F S x₀ | def | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | EquicontinuousOn | A family `F : ι → X → α` of functions from a topological space to a uniform space is
*equicontinuous on `S : Set X`* if it is equicontinuous *within `S`* at each point of `S`. |
protected Set.EquicontinuousOn (H : Set <| X → α) (S : Set X) : Prop :=
EquicontinuousOn ((↑) : H → X → α) S | abbrev | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.EquicontinuousOn | We say that a set `H : Set (X → α)` of functions is equicontinuous on a subset if the family
`(↑) : ↥H → (X → α)` is equicontinuous on that subset. |
UniformEquicontinuous (F : ι → β → α) : Prop :=
∀ U ∈ 𝓤 α, ∀ᶠ xy : β × β in 𝓤 β, ∀ i, (F i xy.1, F i xy.2) ∈ U | def | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | UniformEquicontinuous | A family `F : ι → β → α` of functions between uniform spaces is *uniformly equicontinuous* if,
for all entourages `U ∈ 𝓤 α`, there is an entourage `V ∈ 𝓤 β` such that, whenever `x` and `y` are
`V`-close, we have that, *for all `i : ι`*, `F i x` is `U`-close to `F i y`. |
protected Set.UniformEquicontinuous (H : Set <| β → α) : Prop :=
UniformEquicontinuous ((↑) : H → β → α) | abbrev | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.UniformEquicontinuous | We say that a set `H : Set (X → α)` of functions is uniformly equicontinuous if the family
`(↑) : ↥H → (X → α)` is uniformly equicontinuous. |
UniformEquicontinuousOn (F : ι → β → α) (S : Set β) : Prop :=
∀ U ∈ 𝓤 α, ∀ᶠ xy : β × β in 𝓤 β ⊓ 𝓟 (S ×ˢ S), ∀ i, (F i xy.1, F i xy.2) ∈ U | def | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | UniformEquicontinuousOn | A family `F : ι → β → α` of functions between uniform spaces is
*uniformly equicontinuous on `S : Set β`* if, for all entourages `U ∈ 𝓤 α`, there is a relative
entourage `V ∈ 𝓤 β ⊓ 𝓟 (S ×ˢ S)` such that, whenever `x` and `y` are `V`-close, we have that,
*for all `i : ι`*, `F i x` is `U`-close to `F i y`. |
protected Set.UniformEquicontinuousOn (H : Set <| β → α) (S : Set β) : Prop :=
UniformEquicontinuousOn ((↑) : H → β → α) S | abbrev | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Set.UniformEquicontinuousOn | We say that a set `H : Set (X → α)` of functions is uniformly equicontinuous on a subset if the
family `(↑) : ↥H → (X → α)` is uniformly equicontinuous on that subset. |
EquicontinuousAt.equicontinuousWithinAt {F : ι → X → α} {x₀ : X} (H : EquicontinuousAt F x₀)
(S : Set X) : EquicontinuousWithinAt F S x₀ :=
fun U hU ↦ (H U hU).filter_mono inf_le_left | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | EquicontinuousAt.equicontinuousWithinAt | null |
EquicontinuousWithinAt.mono {F : ι → X → α} {x₀ : X} {S T : Set X}
(H : EquicontinuousWithinAt F T x₀) (hST : S ⊆ T) : EquicontinuousWithinAt F S x₀ :=
fun U hU ↦ (H U hU).filter_mono <| nhdsWithin_mono x₀ hST
@[simp] lemma equicontinuousWithinAt_univ (F : ι → X → α) (x₀ : X) :
EquicontinuousWithinAt F univ x₀ ↔ EquicontinuousAt F x₀ := by
rw [EquicontinuousWithinAt, EquicontinuousAt, nhdsWithin_univ] | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | EquicontinuousWithinAt.mono | null |
equicontinuousAt_restrict_iff (F : ι → X → α) {S : Set X} (x₀ : S) :
EquicontinuousAt (S.restrict ∘ F) x₀ ↔ EquicontinuousWithinAt F S x₀ := by
simp [EquicontinuousWithinAt, EquicontinuousAt,
← eventually_nhds_subtype_iff] | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousAt_restrict_iff | null |
Equicontinuous.equicontinuousOn {F : ι → X → α} (H : Equicontinuous F)
(S : Set X) : EquicontinuousOn F S :=
fun x _ ↦ (H x).equicontinuousWithinAt S | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | Equicontinuous.equicontinuousOn | null |
EquicontinuousOn.mono {F : ι → X → α} {S T : Set X}
(H : EquicontinuousOn F T) (hST : S ⊆ T) : EquicontinuousOn F S :=
fun x hx ↦ (H x (hST hx)).mono hST | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | EquicontinuousOn.mono | null |
equicontinuousOn_univ (F : ι → X → α) :
EquicontinuousOn F univ ↔ Equicontinuous F := by
simp [EquicontinuousOn, Equicontinuous] | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuousOn_univ | null |
equicontinuous_restrict_iff (F : ι → X → α) {S : Set X} :
Equicontinuous (S.restrict ∘ F) ↔ EquicontinuousOn F S := by
simp [Equicontinuous, EquicontinuousOn, equicontinuousAt_restrict_iff] | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | equicontinuous_restrict_iff | null |
UniformEquicontinuous.uniformEquicontinuousOn {F : ι → β → α} (H : UniformEquicontinuous F)
(S : Set β) : UniformEquicontinuousOn F S :=
fun U hU ↦ (H U hU).filter_mono inf_le_left | lemma | Topology | [
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology"
] | Mathlib/Topology/UniformSpace/Equicontinuity.lean | UniformEquicontinuous.uniformEquicontinuousOn | null |
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