statement stringlengths 1 2.93k | proof stringlengths 0 19.2k | type stringclasses 13
values | symbolic_name stringlengths 1 131 | library stringlengths 4 62 | filename stringlengths 20 95 | imports listlengths 0 10 | deps listlengths 0 64 | docstring stringlengths 0 4.95k | source_url stringclasses 1
value | commit stringclasses 1
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uniformity_eq_symm : 𝓤 α = map Prod.swap (𝓤 α) | le_antisymm uniformity_le_symm symm_le_uniformity | theorem | uniformity_eq_symm | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"le_antisymm",
"symm_le_uniformity",
"uniformity_le_symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comap_swap_uniformity : comap (@Prod.swap α α) (𝓤 α) = 𝓤 α | (congr_arg _ uniformity_eq_symm).trans <| comap_map Prod.swap_injective | theorem | comap_swap_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Prod.swap_injective",
"comap",
"trans",
"uniformity_eq_symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
symmetrize_mem_uniformity {V : SetRel α α} (h : V ∈ 𝓤 α) : SetRel.symmetrize V ∈ 𝓤 α | by
apply (𝓤 α).inter_sets h
rw [← comap_swap_uniformity]
exact preimage_mem_comap h | theorem | symmetrize_mem_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"SetRel",
"SetRel.symmetrize",
"comap_swap_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.hasBasis_symmetric :
(𝓤 α).HasBasis (fun s : SetRel α α => s ∈ 𝓤 α ∧ SetRel.IsSymm s) id | hasBasis_self.2 fun t t_in =>
⟨SetRel.symmetrize t, symmetrize_mem_uniformity t_in, inferInstance,
SetRel.symmetrize_subset_self⟩ | theorem | UniformSpace.hasBasis_symmetric | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"SetRel",
"SetRel.IsSymm",
"symmetrize_mem_uniformity"
] | Symmetric entourages form a basis of `𝓤 α` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformity_lift_le_swap {g : SetRel 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 | uniformity_lift_le_swap | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Filter",
"Filter.map",
"Monotone",
"SetRel",
"le_rfl",
"uniformity_le_symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformity_lift_le_comp {f : SetRel 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 : SetRel α α => s ○ s).lift f := by
rw [lift_lift'_assoc]
· exact monotone_id.relComp monotone_id
· exact h
_ ≤ (𝓤 α).lift f := lift_mono comp_le_uniformity le_rfl | theorem | uniformity_lift_le_comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Filter",
"Monotone",
"SetRel",
"comp_le_uniformity",
"le_rfl",
"monotone_id"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp3_mem_uniformity {s : SetRel 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'
have := isRefl_of_mem_uniformity ht
⟨t, ht, (SetRel.comp_subset_comp (SetRel.left_subset_comp.trans htt') htt').trans ht's⟩ | theorem | comp3_mem_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"SetRel",
"SetRel.comp_subset_comp",
"comp_mem_uniformity_sets",
"isRefl_of_mem_uniformity",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_le_uniformity3 : ((𝓤 α).lift' fun s : SetRel 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 | comp_le_uniformity3 | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"SetRel",
"comp3_mem_uniformity"
] | See also `comp3_mem_uniformity`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
comp_symm_mem_uniformity_sets {s : SetRel α α} (hs : s ∈ 𝓤 α) :
∃ t ∈ 𝓤 α, SetRel.IsSymm t ∧ t ○ t ⊆ s | by
obtain ⟨w, w_in, w_sub⟩ : ∃ w ∈ 𝓤 α, w ○ w ⊆ s := comp_mem_uniformity_sets hs
use SetRel.symmetrize w, symmetrize_mem_uniformity w_in, inferInstance
have : SetRel.symmetrize w ⊆ w := SetRel.symmetrize_subset_self
calc SetRel.symmetrize w ○ SetRel.symmetrize w
_ ⊆ w ○ w := by gcongr
_ ⊆ s := w_sub | theorem | comp_symm_mem_uniformity_sets | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"SetRel",
"SetRel.IsSymm",
"SetRel.symmetrize",
"SetRel.symmetrize_subset_self",
"comp_mem_uniformity_sets",
"symmetrize_mem_uniformity"
] | See also `comp_open_symm_mem_uniformity_sets`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
subset_comp_self_of_mem_uniformity {s : SetRel α α} (h : s ∈ 𝓤 α) : s ⊆ s ○ s | have := isRefl_of_mem_uniformity h; SetRel.left_subset_comp | theorem | subset_comp_self_of_mem_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"SetRel",
"SetRel.left_subset_comp",
"isRefl_of_mem_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
comp_comp_symm_mem_uniformity_sets {s : SetRel α α} (hs : s ∈ 𝓤 α) :
∃ t ∈ 𝓤 α, SetRel.IsSymm 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_su... | theorem | comp_comp_symm_mem_uniformity_sets | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"SetRel",
"SetRel.IsSymm",
"comp_symm_mem_uniformity_sets",
"subset_comp_self_of_mem_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball (x : β) (V : Set (β × β)) : Set β | Prod.mk x ⁻¹' V | def | UniformSpace.ball | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set"
] | 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 }`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
mem_ball_self (x : α) {V : SetRel α α} : V ∈ 𝓤 α → x ∈ ball x V | refl_mem_uniformity | lemma | UniformSpace.mem_ball_self | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"SetRel",
"refl_mem_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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) | SetRel.prodMk_mem_comp h h' | theorem | UniformSpace.mem_ball_comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set",
"SetRel.prodMk_mem_comp"
] | The triangle inequality for `UniformSpace.ball` | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | UniformSpace.ball_subset_of_comp_subset | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_mono {V W : Set (β × β)} (h : V ⊆ W) (x : β) : ball x V ⊆ ball x W | preimage_mono h | theorem | UniformSpace.ball_mono | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_inter (x : β) (V W : Set (β × β)) : ball x (V ∩ W) = ball x V ∩ ball x W | preimage_inter | theorem | UniformSpace.ball_inter | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_inter_left (x : β) (V W : Set (β × β)) : ball x (V ∩ W) ⊆ ball x V | ball_mono inter_subset_left x | theorem | UniformSpace.ball_inter_left | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_inter_right (x : β) (V W : Set (β × β)) : ball x (V ∩ W) ⊆ ball x W | ball_mono inter_subset_right x | theorem | UniformSpace.ball_inter_right | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_iInter {x : β} {V : ι → Set (β × β)} : ball x (⋂ i, V i) = ⋂ i, ball x (V i) | preimage_iInter | theorem | UniformSpace.ball_iInter | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_ball_symmetry {V : SetRel β β} [V.IsSymm] {x y} : x ∈ ball y V ↔ y ∈ ball x V | V.comm | theorem | UniformSpace.mem_ball_symmetry | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"SetRel"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
ball_eq_of_symmetry {V : SetRel β β} [V.IsSymm] {x} : ball x V = { y | (y, x) ∈ V } | by
ext y
rw [mem_ball_symmetry]
exact Iff.rfl | theorem | UniformSpace.ball_eq_of_symmetry | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"SetRel"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_comp_of_mem_ball {V W : SetRel β β} {x y z : β} [V.IsSymm] (hx : x ∈ ball z V)
(hy : y ∈ ball z W) : (x, y) ∈ V ○ W | by
rw [mem_ball_symmetry] at hx
exact ⟨z, hx, hy⟩ | theorem | UniformSpace.mem_comp_of_mem_ball | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"SetRel"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_comp_comp {V W M : SetRel β β} [W.IsSymm] {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]⟩, hwz⟩
· rintro ⟨⟨w, z⟩, ⟨w_in, z_in⟩, hwz⟩
rw [mem_ball_symmetry] at z_in
exact ⟨z, ⟨w, w_in, hwz⟩, z_in⟩ | theorem | UniformSpace.mem_comp_comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"SetRel"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
isCover_iff_subset_iUnion_ball {U : SetRel β β} [U.IsSymm] {s N : Set β} :
U.IsCover s N ↔ s ⊆ ⋃ y ∈ N, ball y U | by
simp [SetRel.IsCover, subset_def, ball, U.comm]
alias ⟨_root_.SetRel.IsCover.subset_iUnion_ball, _root_.SetRel.IsCover.of_subset_iUnion_ball⟩ :=
isCover_iff_subset_iUnion_ball | lemma | UniformSpace.isCover_iff_subset_iUnion_ball | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set",
"SetRel",
"SetRel.IsCover"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | mem_nhds_uniformity_iff_right | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set",
"nhds_eq_comap_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | mem_nhds_uniformity_iff_left | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set",
"mem_nhds_uniformity_iff_right",
"uniformity_eq_symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | nhdsWithin_eq_comap_uniformity_of_mem | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set",
"comap",
"nhdsWithin",
"nhds_eq_comap_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | nhdsWithin_eq_comap_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set",
"comap",
"nhdsWithin_eq_comap_uniformity_of_mem"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | isOpen_iff_ball_subset | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"IsOpen",
"Set",
"isOpen_iff_mem_nhds",
"nhds_eq_comap_uniformity"
] | See also `isOpen_iff_isOpen_ball_subset`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
nhds_basis_uniformity' {p : ι → Prop} {s : ι → SetRel 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 | nhds_basis_uniformity' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"SetRel",
"nhds_eq_comap_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_basis_uniformity {p : ι → Prop} {s : ι → SetRel 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 | nhds_basis_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"SetRel",
"comap_swap_uniformity",
"nhds_basis_uniformity'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | nhds_eq_comap_uniformity' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"comap",
"nhds_basis_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | UniformSpace.mem_nhds_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set",
"nhds_eq_comap_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.ball_mem_nhds (x : α) ⦃V : SetRel 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 | UniformSpace.ball_mem_nhds | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"SetRel",
"UniformSpace.mem_nhds_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.ball_mem_nhdsWithin {x : α} {S : Set α} ⦃V : SetRel 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 | UniformSpace.ball_mem_nhdsWithin | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set",
"SetRel",
"nhdsWithin_eq_comap_uniformity_of_mem"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.mem_nhds_iff_symm {x : α} {s : Set α} :
s ∈ 𝓝 x ↔ ∃ V ∈ 𝓤 α, SetRel.IsSymm V ∧ ball x V ⊆ s | by
rw [UniformSpace.mem_nhds_iff]
constructor
· rintro ⟨V, V_in, V_sub⟩
use SetRel.symmetrize V, symmetrize_mem_uniformity V_in, inferInstance
exact Subset.trans (ball_mono SetRel.symmetrize_subset_self x) V_sub
· rintro ⟨V, V_in, _, V_sub⟩
exact ⟨V, V_in, V_sub⟩ | theorem | UniformSpace.mem_nhds_iff_symm | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set",
"SetRel.IsSymm",
"SetRel.symmetrize",
"SetRel.symmetrize_subset_self",
"UniformSpace.mem_nhds_iff",
"symmetrize_mem_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.hasBasis_nhds (x : α) :
HasBasis (𝓝 x) (fun s : SetRel α α => s ∈ 𝓤 α ∧ SetRel.IsSymm s) fun s => ball x s | ⟨fun t => by simp [UniformSpace.mem_nhds_iff_symm, and_assoc]⟩ | theorem | UniformSpace.hasBasis_nhds | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"SetRel",
"SetRel.IsSymm",
"UniformSpace.mem_nhds_iff_symm"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.mem_closure_iff_symm_ball {s : Set α} {x} :
x ∈ closure s ↔ ∀ {V}, V ∈ 𝓤 α → SetRel.IsSymm V → (s ∩ ball x V).Nonempty | by
simp [mem_closure_iff_nhds_basis (hasBasis_nhds x), Set.Nonempty] | theorem | UniformSpace.mem_closure_iff_symm_ball | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set",
"Set.Nonempty",
"SetRel.IsSymm",
"closure",
"mem_closure_iff_nhds_basis"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | UniformSpace.mem_closure_iff_ball | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set",
"closure",
"mem_closure_iff_nhds_basis'",
"nhds_basis_uniformity'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.closure_subset_preimage
{U : SetRel α α} (hU : U ∈ 𝓤 α) (s : Set α) : closure s ⊆ U.preimage s | by
intro x hx
obtain ⟨y, hxy, hy⟩ := mem_closure_iff_ball.mp hx hU
exact ⟨y, hy, hxy⟩ | theorem | UniformSpace.closure_subset_preimage | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set",
"SetRel",
"closure"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformSpace.closure_subset_image
{U : SetRel α α} (hU : U ∈ 𝓤 α) (s : Set α) : closure s ⊆ U.image s | closure_subset_preimage (symm_le_uniformity hU) s | theorem | UniformSpace.closure_subset_image | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set",
"SetRel",
"closure",
"symm_le_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_eq_uniformity {x : α} : 𝓝 x = (𝓤 α).lift' (ball x) | (nhds_basis_uniformity' (𝓤 α).basis_sets).eq_biInf | theorem | nhds_eq_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"nhds_basis_uniformity'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | nhds_eq_uniformity' | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"nhds_basis_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_nhds_left (x : α) {s : SetRel α α} (h : s ∈ 𝓤 α) : { y : α | (x, y) ∈ s } ∈ 𝓝 x | ball_mem_nhds x h | theorem | mem_nhds_left | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"SetRel"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
mem_nhds_right (y : α) {s : SetRel α α} (h : s ∈ 𝓤 α) : { x : α | (x, y) ∈ s } ∈ 𝓝 y | mem_nhds_left _ (symm_le_uniformity h) | theorem | mem_nhds_right | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"SetRel",
"mem_nhds_left",
"symm_le_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | exists_mem_nhds_ball_subset_of_mem_nhds | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set",
"UniformSpace.ball",
"comp_mem_uniformity_sets",
"mem_nhds_left"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_right_nhds_uniformity {a : α} : Tendsto (fun a' => (a', a)) (𝓝 a) (𝓤 α) | fun _ =>
mem_nhds_right a | theorem | tendsto_right_nhds_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"mem_nhds_right"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
tendsto_left_nhds_uniformity {a : α} : Tendsto (fun a' => (a, a')) (𝓝 a) (𝓤 α) | fun _ =>
mem_nhds_left a | theorem | tendsto_left_nhds_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"mem_nhds_left"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lift_nhds_left {x : α} {g : Set α → Filter β} (hg : Monotone g) :
(𝓝 x).lift g = (𝓤 α).lift fun s : SetRel α α => g (ball x s) | by
rw [nhds_eq_comap_uniformity, comap_lift_eq2 hg]
simp_rw [ball, Function.comp_def] | theorem | lift_nhds_left | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Filter",
"Monotone",
"Set",
"SetRel",
"nhds_eq_comap_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
lift_nhds_right {x : α} {g : Set α → Filter β} (hg : Monotone g) :
(𝓝 x).lift g = (𝓤 α).lift fun s : SetRel α α => g { y | (y, x) ∈ s } | by
rw [nhds_eq_comap_uniformity', comap_lift_eq2 hg]
simp_rw [Function.comp_def, preimage] | theorem | lift_nhds_right | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Filter",
"Monotone",
"Set",
"SetRel",
"nhds_eq_comap_uniformity'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
nhds_nhds_eq_uniformity_uniformity_prod {a b : α} :
𝓝 a ×ˢ 𝓝 b = (𝓤 α).lift fun s : SetRel 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 | nhds_nhds_eq_uniformity_uniformity_prod | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"SetRel",
"nhds_eq_uniformity",
"nhds_eq_uniformity'"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.biInter_biUnion_ball {p : ι → Prop} {U : ι → SetRel 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] | theorem | Filter.HasBasis.biInter_biUnion_ball | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set",
"SetRel",
"closure",
"mem_closure_iff_nhds_basis",
"nhds_basis_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuous (f : α → β) | Tendsto (fun x : α × α => (f x.1, f x.2)) (𝓤 α) (𝓤 β)
/-- Notation for uniform continuity with respect to non-standard `UniformSpace` instances. -/
scoped[Uniformity] notation "UniformContinuous[" u₁ ", " u₂ "]" => @UniformContinuous _ _ u₁ u₂ | def | UniformContinuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
UniformContinuousOn (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 | UniformContinuousOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
uniformContinuous_def {f : α → β} :
UniformContinuous f ↔ ∀ r ∈ 𝓤 β, { x : α × α | (f x.1, f x.2) ∈ r } ∈ 𝓤 α | Iff.rfl | theorem | uniformContinuous_def | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_iff_eventually {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 | uniformContinuous_iff_eventually | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuousOn_univ {f : α → β} :
UniformContinuousOn f univ ↔ UniformContinuous f | by
rw [UniformContinuousOn, UniformContinuous, univ_prod_univ, principal_univ, inf_top_eq] | theorem | uniformContinuousOn_univ | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"UniformContinuous",
"UniformContinuousOn"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_of_const {c : α → β} (h : ∀ a b, c a = c b) :
UniformContinuous c | have : (fun x : α × α => (c x.fst, c x.snd)) ⁻¹' SetRel.id = 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 | uniformContinuous_of_const | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"SetRel.id",
"UniformContinuous",
"le_trans",
"refl_le_uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_id : UniformContinuous (@id α) | tendsto_id | theorem | uniformContinuous_id | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"UniformContinuous"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous_const {b : β} : UniformContinuous fun _ : α => b | uniformContinuous_of_const fun _ _ => rfl | theorem | uniformContinuous_const | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"UniformContinuous",
"uniformContinuous_of_const"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuous.comp [UniformSpace γ] {g : β → γ} {f : α → β}
(hg : UniformContinuous g) (hf : UniformContinuous f) : UniformContinuous (g ∘ f) | hg.comp hf | theorem | UniformContinuous.comp | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"UniformContinuous",
"UniformSpace"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
UniformContinuous.iterate (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 | UniformContinuous.iterate | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Function.iterate_succ",
"UniformContinuous",
"UniformContinuous.comp",
"uniformContinuous_id"
] | If a function `T` is uniformly continuous in a uniform space `β`,
then its `n`-th iterate `T^[n]` is also uniformly continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Filter.HasBasis.uniformContinuous_iff {ι'} {p : ι → Prop}
{s : ι → SetRel 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 | Filter.HasBasis.uniformContinuous_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set",
"SetRel",
"UniformContinuous",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
Filter.HasBasis.uniformContinuousOn_iff {ι'} {p : ι → Prop}
{s : ι → SetRel 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) ∈ ... | ((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 | Filter.HasBasis.uniformContinuousOn_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"Set",
"Set.inter_comm",
"SetRel",
"UniformContinuousOn",
"forall_mem_comm",
"trans"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUniformInducing (f : α → β) : Prop where
/-- The uniformity filter on the domain is the pullback of the uniformity filter on the codomain
under `Prod.map f f`. -/
comap_uniformity : comap (fun x : α × α ↦ (f x.1, f x.2)) (𝓤 β) = 𝓤 α | structure | IsUniformInducing | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"comap"
] | A map `f : α → β` between uniform spaces is called *uniform inducing* if the uniformity filter
on `α` is the pullback of the uniformity filter on `β` under `Prod.map f f`. If `α` is a separated
space, then this implies that `f` is injective, hence it is a `IsUniformEmbedding`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUniformEmbedding (f : α → β) : Prop extends IsUniformInducing f where
/-- A uniform embedding is injective. -/
injective : Function.Injective f | structure | IsUniformEmbedding | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"IsUniformInducing"
] | A map `f : α → β` between uniform spaces is a *uniform embedding* if it is uniform inducing and
injective. If `α` is a separated space, then the latter assumption follows from the former. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
IsUniformEmbedding.isUniformInducing {f : α → β} (hf : IsUniformEmbedding f) :
IsUniformInducing f | hf.toIsUniformInducing | lemma | IsUniformEmbedding.isUniformInducing | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Defs.lean | [] | [
"IsUniformEmbedding",
"IsUniformInducing"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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_... | lemma | Monotone.tendstoLocallyUniformly_of_forall_tendsto | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Dini.lean | [] | [
"Continuous",
"Filter",
"Metric.tendstoLocallyUniformly_iff",
"Monotone",
"TendstoLocallyUniformly",
"continuous_const",
"norm_le_norm_of_abs_le_abs",
"tendstoLocallyUniformly_iff_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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Monotone.tendstoLocallyUniformlyOn_of_forall_tendsto | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Dini.lean | [] | [
"ContinuousOn",
"Monotone",
"Set",
"TendstoLocallyUniformlyOn",
"tendstoLocallyUniformlyOn_iff_tendstoLocallyUniformly_comp_coe"
] | **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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Monotone.tendstoUniformly_of_forall_tendsto | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Dini.lean | [] | [
"CompactSpace",
"Continuous",
"Monotone",
"TendstoUniformly"
] | **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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Monotone.tendstoUniformlyOn_of_forall_tendsto | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Dini.lean | [] | [
"ContinuousOn",
"IsCompact",
"Monotone",
"Set",
"TendstoUniformlyOn",
"tendstoLocallyUniformlyOn_iff_tendstoUniformlyOn_of_compact"
] | **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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Antitone.tendstoLocallyUniformly_of_forall_tendsto | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Dini.lean | [] | [
"Antitone",
"Continuous",
"Monotone.tendstoLocallyUniformly_of_forall_tendsto",
"TendstoLocallyUniformly"
] | **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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Antitone.tendstoLocallyUniformlyOn_of_forall_tendsto | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Dini.lean | [] | [
"Antitone",
"ContinuousOn",
"Monotone.tendstoLocallyUniformlyOn_of_forall_tendsto",
"Set",
"TendstoLocallyUniformlyOn"
] | **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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Antitone.tendstoUniformly_of_forall_tendsto | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Dini.lean | [] | [
"Antitone",
"CompactSpace",
"Continuous",
"Monotone.tendstoUniformly_of_forall_tendsto",
"TendstoUniformly"
] | **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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | Antitone.tendstoUniformlyOn_of_forall_tendsto | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Dini.lean | [] | [
"Antitone",
"ContinuousOn",
"IsCompact",
"Monotone.tendstoUniformlyOn_of_forall_tendsto",
"Set",
"TendstoUniformlyOn"
] | **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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | ContinuousMap.tendsto_of_monotone_of_pointwise | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Dini.lean | [] | [
"Monotone"
] | **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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | ContinuousMap.tendsto_of_antitone_of_pointwise | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Dini.lean | [] | [
"Antitone"
] | **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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
DiscreteUniformity (X : Type*) [u : UniformSpace X] : Prop where
eq_bot : u = ⊥ | class | DiscreteUniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/DiscreteUniformity.lean | [] | [
"UniformSpace"
] | The discrete uniformity | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.discreteUniformity_iff_eq_principal_setRelId {X : Type*} [UniformSpace X] :
DiscreteUniformity X ↔ uniformity X = 𝓟 SetRel.id | by
rw [discreteUniformity_iff_eq_bot, UniformSpace.ext_iff, Filter.ext_iff, bot_uniformity]
@[deprecated (since := "2025-12-19")]
alias _root_.discreteUniformity_iff_eq_principal_relId :=
_root_.discreteUniformity_iff_eq_principal_setRelId | theorem | discreteUniformity_iff_eq_principal_setRelId | Topology.UniformSpace | Mathlib/Topology/UniformSpace/DiscreteUniformity.lean | [] | [
"DiscreteUniformity",
"SetRel.id",
"UniformSpace",
"UniformSpace.ext_iff",
"bot_uniformity",
"uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
eq_principal_setRelId : uniformity X = 𝓟 SetRel.id | discreteUniformity_iff_eq_principal_setRelId.mp inferInstance
@[deprecated (since := "2025-12-19")] alias eq_principal_relId := eq_principal_setRelId | theorem | DiscreteUniformity.eq_principal_setRelId | Topology.UniformSpace | Mathlib/Topology/UniformSpace/DiscreteUniformity.lean | [] | [
"SetRel.id",
"uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
_root_.discreteUniformity_iff_setRelId_mem_uniformity {X : Type*} [UniformSpace X] :
DiscreteUniformity X ↔ SetRel.id ∈ uniformity X | by
rw [← uniformSpace_eq_bot, discreteUniformity_iff_eq_bot]
@[deprecated (since := "2025-12-19")]
alias _root_.discreteUniformity_iff_relId_mem_uniformity :=
_root_.discreteUniformity_iff_setRelId_mem_uniformity | theorem | discreteUniformity_iff_setRelId_mem_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/DiscreteUniformity.lean | [] | [
"DiscreteUniformity",
"SetRel.id",
"UniformSpace",
"uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
relId_mem_uniformity : SetRel.id ∈ uniformity X | discreteUniformity_iff_setRelId_mem_uniformity.mp inferInstance | theorem | DiscreteUniformity.relId_mem_uniformity | Topology.UniformSpace | Mathlib/Topology/UniformSpace/DiscreteUniformity.lean | [] | [
"SetRel.id",
"uniformity"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
uniformContinuous {Y : Type*} [UniformSpace Y] (f : X → Y) :
UniformContinuous f | by
simp only [uniformContinuous_iff_le_comap, DiscreteUniformity.eq_bot, bot_le] | theorem | DiscreteUniformity.uniformContinuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/DiscreteUniformity.lean | [] | [
"UniformContinuous",
"UniformSpace",
"uniformContinuous_iff_le_comap"
] | On a space with a discrete uniformity, any function is uniformly continuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | EquicontinuousAt | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.EquicontinuousAt (H : Set <| X → α) (x₀ : X) : Prop | EquicontinuousAt ((↑) : H → X → α) x₀ | abbrev | Set.EquicontinuousAt | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousAt",
"Set"
] | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | EquicontinuousWithinAt | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.EquicontinuousWithinAt (H : Set <| X → α) (S : Set X) (x₀ : X) : Prop | EquicontinuousWithinAt ((↑) : H → X → α) S x₀ | abbrev | Set.EquicontinuousWithinAt | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousWithinAt",
"Set"
] | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Equicontinuous (F : ι → X → α) : Prop | ∀ x₀, EquicontinuousAt F x₀ | def | Equicontinuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousAt"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.Equicontinuous (H : Set <| X → α) : Prop | Equicontinuous ((↑) : H → X → α) | abbrev | Set.Equicontinuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Equicontinuous",
"Set"
] | We say that a set `H : Set (X → α)` of functions is equicontinuous if the family
`(↑) : ↥H → (X → α)` is equicontinuous. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
EquicontinuousOn (F : ι → X → α) (S : Set X) : Prop | ∀ x₀ ∈ S, EquicontinuousWithinAt F S x₀ | def | EquicontinuousOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousWithinAt",
"Set"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.EquicontinuousOn (H : Set <| X → α) (S : Set X) : Prop | EquicontinuousOn ((↑) : H → X → α) S | abbrev | Set.EquicontinuousOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousOn",
"Set"
] | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | UniformEquicontinuous | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.UniformEquicontinuous (H : Set <| β → α) : Prop | UniformEquicontinuous ((↑) : H → β → α) | abbrev | Set.UniformEquicontinuous | Topology.UniformSpace | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | UniformEquicontinuousOn | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"Set"
] | 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`. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
Set.UniformEquicontinuousOn (H : Set <| β → α) (S : Set β) : Prop | UniformEquicontinuousOn ((↑) : H → β → α) S | abbrev | Set.UniformEquicontinuousOn | Topology.UniformSpace | 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. | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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 | EquicontinuousAt.equicontinuousWithinAt | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousAt",
"EquicontinuousWithinAt",
"Set"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | lemma | EquicontinuousWithinAt.mono | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousWithinAt",
"Set",
"nhdsWithin_mono"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
equicontinuousWithinAt_univ (F : ι → X → α) (x₀ : X) :
EquicontinuousWithinAt F univ x₀ ↔ EquicontinuousAt F x₀ | by
rw [EquicontinuousWithinAt, EquicontinuousAt, nhdsWithin_univ] | lemma | equicontinuousWithinAt_univ | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousAt",
"EquicontinuousWithinAt",
"nhdsWithin_univ"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 | |
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 | equicontinuousAt_restrict_iff | Topology.UniformSpace | Mathlib/Topology/UniformSpace/Equicontinuity.lean | [] | [
"EquicontinuousAt",
"EquicontinuousWithinAt",
"Set",
"eventually_nhds_subtype_iff"
] | https://github.com/leanprover-community/mathlib4 | b9f14353520df73472ae3825fb53f86559a01319 |
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