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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UniformCompletion.completeEquivSelf [CompleteSpace α] [T0Space α] : Completion α ≃ᵤ α :=
AbstractCompletion.compareEquiv Completion.cPkg AbstractCompletion.ofComplete
open TopologicalSpace | def | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | UniformCompletion.completeEquivSelf | The uniform bijection between a complete space and its uniform completion. |
separableSpace_completion [SeparableSpace α] : SeparableSpace (Completion α) :=
Completion.isDenseInducing_coe.separableSpace | instance | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | separableSpace_completion | null |
isDenseEmbedding_coe [T0Space α] : IsDenseEmbedding ((↑) : α → Completion α) :=
{ isDenseInducing_coe with injective := separated_pureCauchy_injective } | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | isDenseEmbedding_coe | null |
denseRange_coe₂ :
DenseRange fun x : α × β => ((x.1 : Completion α), (x.2 : Completion β)) :=
denseRange_coe.prodMap denseRange_coe | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | denseRange_coe₂ | null |
denseRange_coe₃ :
DenseRange fun x : α × β × γ =>
((x.1 : Completion α), ((x.2.1 : Completion β), (x.2.2 : Completion γ))) :=
denseRange_coe.prodMap denseRange_coe₂
@[elab_as_elim] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | denseRange_coe₃ | null |
induction_on {p : Completion α → Prop} (a : Completion α) (hp : IsClosed { a | p a })
(ih : ∀ a : α, p a) : p a :=
isClosed_property denseRange_coe hp ih a
@[elab_as_elim] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | induction_on | null |
induction_on₂ {p : Completion α → Completion β → Prop} (a : Completion α) (b : Completion β)
(hp : IsClosed { x : Completion α × Completion β | p x.1 x.2 })
(ih : ∀ (a : α) (b : β), p a b) : p a b :=
have : ∀ x : Completion α × Completion β, p x.1 x.2 :=
isClosed_property denseRange_coe₂ hp fun ⟨a, b⟩ => ih a b
this (a, b)
@[elab_as_elim] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | induction_on₂ | null |
induction_on₃ {p : Completion α → Completion β → Completion γ → Prop} (a : Completion α)
(b : Completion β) (c : Completion γ)
(hp : IsClosed { x : Completion α × Completion β × Completion γ | p x.1 x.2.1 x.2.2 })
(ih : ∀ (a : α) (b : β) (c : γ), p a b c) : p a b c :=
have : ∀ x : Completion α × Completion β × Completion γ, p x.1 x.2.1 x.2.2 :=
isClosed_property denseRange_coe₃ hp fun ⟨a, b, c⟩ => ih a b c
this (a, b, c) | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | induction_on₃ | null |
ext {Y : Type*} [TopologicalSpace Y] [T2Space Y] {f g : Completion α → Y}
(hf : Continuous f) (hg : Continuous g) (h : ∀ a : α, f a = g a) : f = g :=
cPkg.funext hf hg h | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | ext | null |
ext' {Y : Type*} [TopologicalSpace Y] [T2Space Y] {f g : Completion α → Y}
(hf : Continuous f) (hg : Continuous g) (h : ∀ a : α, f a = g a) (a : Completion α) :
f a = g a :=
congr_fun (ext hf hg h) a | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | ext' | null |
protected extension (f : α → β) : Completion α → β :=
cPkg.extend f | def | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | extension | "Extension" to the completion. It is defined for any map `f` but
returns an arbitrary constant value if `f` is not uniformly continuous |
uniformContinuous_extension : UniformContinuous (Completion.extension f) :=
cPkg.uniformContinuous_extend
@[continuity, fun_prop] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | uniformContinuous_extension | null |
continuous_extension : Continuous (Completion.extension f) :=
cPkg.continuous_extend | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | continuous_extension | null |
extension_coe [T0Space β] (hf : UniformContinuous f) (a : α) :
(Completion.extension f) a = f a :=
cPkg.extend_coe hf a | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | extension_coe | null |
inseparable_extension_coe (hf : UniformContinuous f) (x : α) :
Inseparable (Completion.extension f x) (f x) :=
cPkg.inseparable_extend_coe hf x | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | inseparable_extension_coe | null |
isUniformInducing_extension [CompleteSpace β] (h : IsUniformInducing f) :
IsUniformInducing (Completion.extension f) :=
cPkg.isUniformInducing_extend h
variable [T0Space β] [CompleteSpace β] | lemma | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | isUniformInducing_extension | null |
extension_unique (hf : UniformContinuous f) {g : Completion α → β}
(hg : UniformContinuous g) (h : ∀ a : α, f a = g (a : Completion α)) :
Completion.extension f = g :=
cPkg.extend_unique hf hg h
@[simp] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | extension_unique | null |
extension_comp_coe {f : Completion α → β} (hf : UniformContinuous f) :
Completion.extension (f ∘ (↑)) = f :=
cPkg.extend_comp_coe hf | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | extension_comp_coe | null |
protected map (f : α → β) : Completion α → Completion β :=
cPkg.map cPkg f | def | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | map | Completion functor acting on morphisms |
uniformContinuous_map : UniformContinuous (Completion.map f) :=
cPkg.uniformContinuous_map cPkg f
@[continuity] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | uniformContinuous_map | null |
continuous_map : Continuous (Completion.map f) :=
cPkg.continuous_map cPkg f | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | continuous_map | null |
map_coe (hf : UniformContinuous f) (a : α) : (Completion.map f) a = f a :=
cPkg.map_coe cPkg hf a | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | map_coe | null |
map_unique {f : α → β} {g : Completion α → Completion β} (hg : UniformContinuous g)
(h : ∀ a : α, ↑(f a) = g a) : Completion.map f = g :=
cPkg.map_unique cPkg hg h
@[simp] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | map_unique | null |
map_id : Completion.map (@id α) = id :=
cPkg.map_id | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | map_id | null |
extension_map [CompleteSpace γ] [T0Space γ] {f : β → γ} {g : α → β}
(hf : UniformContinuous f) (hg : UniformContinuous g) :
Completion.extension f ∘ Completion.map g = Completion.extension (f ∘ g) :=
Completion.ext (continuous_extension.comp continuous_map) continuous_extension <| by
simp [hf, hg, hf.comp hg, map_coe, extension_coe] | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | extension_map | null |
map_comp {g : β → γ} {f : α → β} (hg : UniformContinuous g) (hf : UniformContinuous f) :
Completion.map g ∘ Completion.map f = Completion.map (g ∘ f) :=
extension_map ((uniformContinuous_coe _).comp hg) hf | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | map_comp | null |
completionSeparationQuotientEquiv (α : Type u) [UniformSpace α] :
Completion (SeparationQuotient α) ≃ Completion α := by
refine ⟨Completion.extension (lift' ((↑) : α → Completion α)),
Completion.map SeparationQuotient.mk, fun a ↦ ?_, fun a ↦ ?_⟩
· refine induction_on a (isClosed_eq (continuous_map.comp continuous_extension) continuous_id) ?_
refine SeparationQuotient.surjective_mk.forall.2 fun a ↦ ?_
rw [extension_coe (uniformContinuous_lift' _), lift'_mk (uniformContinuous_coe α),
map_coe uniformContinuous_mk]
· refine induction_on a
(isClosed_eq (continuous_extension.comp continuous_map) continuous_id) fun a ↦ ?_
rw [map_coe uniformContinuous_mk, extension_coe (uniformContinuous_lift' _),
lift'_mk (uniformContinuous_coe _)] | def | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | completionSeparationQuotientEquiv | The isomorphism between the completion of a uniform space and the completion of its separation
quotient. |
uniformContinuous_completionSeparationQuotientEquiv :
UniformContinuous (completionSeparationQuotientEquiv α) :=
uniformContinuous_extension | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | uniformContinuous_completionSeparationQuotientEquiv | null |
uniformContinuous_completionSeparationQuotientEquiv_symm :
UniformContinuous (completionSeparationQuotientEquiv α).symm :=
uniformContinuous_map | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | uniformContinuous_completionSeparationQuotientEquiv_symm | null |
protected extension₂ (f : α → β → γ) : Completion α → Completion β → γ :=
cPkg.extend₂ cPkg f | def | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | extension₂ | Extend a two variable map to the Hausdorff completions. |
extension₂_coe_coe (hf : UniformContinuous₂ f) (a : α) (b : β) :
Completion.extension₂ f a b = f a b :=
cPkg.extension₂_coe_coe cPkg hf a b | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | extension₂_coe_coe | null |
uniformContinuous_extension₂ : UniformContinuous₂ (Completion.extension₂ f) :=
cPkg.uniformContinuous_extension₂ cPkg f | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | uniformContinuous_extension₂ | null |
protected map₂ (f : α → β → γ) : Completion α → Completion β → Completion γ :=
cPkg.map₂ cPkg cPkg f | def | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | map₂ | Lift a two variable map to the Hausdorff completions. |
uniformContinuous_map₂ (f : α → β → γ) : UniformContinuous₂ (Completion.map₂ f) :=
cPkg.uniformContinuous_map₂ cPkg cPkg f | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | uniformContinuous_map₂ | null |
continuous_map₂ {δ} [TopologicalSpace δ] {f : α → β → γ} {a : δ → Completion α}
{b : δ → Completion β} (ha : Continuous a) (hb : Continuous b) :
Continuous fun d : δ => Completion.map₂ f (a d) (b d) :=
cPkg.continuous_map₂ cPkg cPkg ha hb | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | continuous_map₂ | null |
map₂_coe_coe (a : α) (b : β) (f : α → β → γ) (hf : UniformContinuous₂ f) :
Completion.map₂ f (a : Completion α) (b : Completion β) = f a b :=
cPkg.map₂_coe_coe cPkg cPkg a b f hf | theorem | Topology | [
"Mathlib.Topology.UniformSpace.AbstractCompletion"
] | Mathlib/Topology/UniformSpace/Completion.lean | map₂_coe_coe | null |
idRel {α : Type*} :=
{ p : α × α | p.1 = p.2 }
@[simp] | def | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | idRel | The identity relation, or the graph of the identity function |
mem_idRel {a b : α} : (a, b) ∈ @idRel α ↔ a = b :=
Iff.rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | mem_idRel | null |
idRel_subset {s : Set (α × α)} : idRel ⊆ s ↔ ∀ a, (a, a) ∈ s := by
simp [subset_def] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | idRel_subset | null |
eq_singleton_left_of_prod_subset_idRel {X : Type*} {S T : Set X} (hS : S.Nonempty)
(hT : T.Nonempty) (h_diag : S ×ˢ T ⊆ idRel) : ∃ x, S = {x} := by
rcases hS, hT with ⟨⟨s, hs⟩, ⟨t, ht⟩⟩
refine ⟨s, eq_singleton_iff_nonempty_unique_mem.mpr ⟨⟨s, hs⟩, fun x hx ↦ ?_⟩⟩
rw [prod_subset_iff] at h_diag
replace hs := h_diag s hs t ht
replace hx := h_diag x hx t ht
simp only [idRel, mem_setOf_eq] at hx hs
rwa [← hs] at hx | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | eq_singleton_left_of_prod_subset_idRel | null |
eq_singleton_right_prod_subset_idRel {X : Type*} {S T : Set X} (hS : S.Nonempty)
(hT : T.Nonempty) (h_diag : S ×ˢ T ⊆ idRel) : ∃ x, T = {x} := by
rw [Set.prod_subset_iff] at h_diag
replace h_diag := fun x hx y hy => (h_diag y hy x hx).symm
exact eq_singleton_left_of_prod_subset_idRel hT hS (prod_subset_iff.mpr h_diag) | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | eq_singleton_right_prod_subset_idRel | null |
eq_singleton_prod_subset_idRel {X : Type*} {S T : Set X} (hS : S.Nonempty)
(hT : T.Nonempty) (h_diag : S ×ˢ T ⊆ idRel) : ∃ x, S = {x} ∧ T = {x} := by
obtain ⟨⟨x, hx⟩, ⟨y, hy⟩⟩ := eq_singleton_left_of_prod_subset_idRel hS hT h_diag,
eq_singleton_right_prod_subset_idRel hS hT h_diag
refine ⟨x, ⟨hx, ?_⟩⟩
rw [hy, Set.singleton_eq_singleton_iff]
exact (Set.prod_subset_iff.mp h_diag x (by simp only [hx, Set.mem_singleton]) y
(by simp only [hy, Set.mem_singleton])).symm | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | eq_singleton_prod_subset_idRel | null |
compRel (r₁ r₂ : Set (α × α)) :=
{ p : α × α | ∃ z : α, (p.1, z) ∈ r₁ ∧ (z, p.2) ∈ r₂ }
@[inherit_doc]
scoped[Uniformity] infixl:62 " ○ " => compRel
open Uniformity
@[simp] | def | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | compRel | The composition of relations |
mem_compRel {α : Type u} {r₁ r₂ : Set (α × α)} {x y : α} :
(x, y) ∈ r₁ ○ r₂ ↔ ∃ z, (x, z) ∈ r₁ ∧ (z, y) ∈ r₂ :=
Iff.rfl
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | mem_compRel | null |
swap_idRel : Prod.swap '' idRel = @idRel α :=
Set.ext fun ⟨a, b⟩ => by simpa [image_swap_eq_preimage_swap] using eq_comm | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | swap_idRel | null |
Monotone.compRel [Preorder β] {f g : β → Set (α × α)} (hf : Monotone f) (hg : Monotone g) :
Monotone fun x => f x ○ g x := fun _ _ h _ ⟨z, h₁, h₂⟩ => ⟨z, hf h h₁, hg h h₂⟩
@[mono, gcongr] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | Monotone.compRel | null |
compRel_mono {f g h k : Set (α × α)} (h₁ : f ⊆ h) (h₂ : g ⊆ k) : f ○ g ⊆ h ○ k :=
fun _ ⟨z, h, h'⟩ => ⟨z, h₁ h, h₂ h'⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | compRel_mono | null |
prodMk_mem_compRel {a b c : α} {s t : Set (α × α)} (h₁ : (a, c) ∈ s) (h₂ : (c, b) ∈ t) :
(a, b) ∈ s ○ t :=
⟨c, h₁, h₂⟩
@[deprecated (since := "2025-03-10")]
alias prod_mk_mem_compRel := prodMk_mem_compRel
@[simp] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | prodMk_mem_compRel | null |
id_compRel {r : Set (α × α)} : idRel ○ r = r :=
Set.ext fun ⟨a, b⟩ => by simp | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | id_compRel | null |
compRel_assoc {r s t : Set (α × α)} : r ○ s ○ t = r ○ (s ○ t) := by
ext ⟨a, b⟩; simp only [mem_compRel]; tauto | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | compRel_assoc | null |
left_subset_compRel {s t : Set (α × α)} (h : idRel ⊆ t) : s ⊆ s ○ t := fun ⟨_x, y⟩ xy_in =>
⟨y, xy_in, h <| rfl⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | left_subset_compRel | null |
right_subset_compRel {s t : Set (α × α)} (h : idRel ⊆ s) : t ⊆ s ○ t := fun ⟨x, _y⟩ xy_in =>
⟨x, h <| rfl, xy_in⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | right_subset_compRel | null |
subset_comp_self {s : Set (α × α)} (h : idRel ⊆ s) : s ⊆ s ○ s :=
left_subset_compRel 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 | null |
subset_iterate_compRel {s t : Set (α × α)} (h : idRel ⊆ s) (n : ℕ) :
t ⊆ (s ○ ·)^[n] t := by
induction n generalizing t with
| zero => exact Subset.rfl
| succ n ihn => exact (right_subset_compRel h).trans ihn | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | subset_iterate_compRel | null |
IsSymmetricRel (V : Set (α × α)) : Prop :=
Prod.swap ⁻¹' V = V
@[deprecated (since := "2025-03-05")]
alias SymmetricRel := IsSymmetricRel | def | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | IsSymmetricRel | The relation is invariant under swapping factors. |
symmetrizeRel (V : Set (α × α)) : Set (α × α) :=
V ∩ Prod.swap ⁻¹' V | def | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | symmetrizeRel | The maximal symmetric relation contained in a given relation. |
symmetric_symmetrizeRel (V : Set (α × α)) : IsSymmetricRel (symmetrizeRel V) := by
simp [IsSymmetricRel, symmetrizeRel, preimage_inter, inter_comm, ← preimage_comp] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | symmetric_symmetrizeRel | null |
symmetrizeRel_subset_self (V : Set (α × α)) : symmetrizeRel V ⊆ V :=
sep_subset _ _
@[mono] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | symmetrizeRel_subset_self | null |
symmetrize_mono {V W : Set (α × α)} (h : V ⊆ W) : symmetrizeRel V ⊆ symmetrizeRel W :=
inter_subset_inter h <| 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 | symmetrize_mono | null |
IsSymmetricRel.mk_mem_comm {V : Set (α × α)} (hV : IsSymmetricRel V) {x y : α} :
(x, y) ∈ V ↔ (y, x) ∈ V :=
Set.ext_iff.1 hV (y, x)
@[deprecated (since := "2025-03-05")]
alias SymmetricRel.mk_mem_comm := IsSymmetricRel.mk_mem_comm | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | IsSymmetricRel.mk_mem_comm | null |
IsSymmetricRel.eq {U : Set (α × α)} (hU : IsSymmetricRel U) : Prod.swap ⁻¹' U = U :=
hU
@[deprecated (since := "2025-03-05")]
alias SymmetricRel.eq := IsSymmetricRel.eq | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | IsSymmetricRel.eq | null |
IsSymmetricRel.inter {U V : Set (α × α)} (hU : IsSymmetricRel U) (hV : IsSymmetricRel V) :
IsSymmetricRel (U ∩ V) := by rw [IsSymmetricRel, preimage_inter, hU.eq, hV.eq]
@[deprecated (since := "2025-03-05")]
alias SymmetricRel.inter := IsSymmetricRel.inter | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | IsSymmetricRel.inter | null |
IsSymmetricRel.iInter {U : (i : ι) → Set (α × α)} (hU : ∀ i, IsSymmetricRel (U i)) :
IsSymmetricRel (⋂ i, U i) := by
simp_rw [IsSymmetricRel, preimage_iInter, (hU _).eq] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | IsSymmetricRel.iInter | null |
IsSymmetricRel.sInter {s : Set (Set (α × α))} (h : ∀ i ∈ s, IsSymmetricRel i) :
IsSymmetricRel (⋂₀ s) := by
rw [sInter_eq_iInter]
exact IsSymmetricRel.iInter (by simpa) | lemma | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | IsSymmetricRel.sInter | null |
isSymmetricRel_idRel : IsSymmetricRel (idRel : Set (α × α)) := by
simp [IsSymmetricRel, idRel, eq_comm] | lemma | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | isSymmetricRel_idRel | null |
isSymmetricRel_univ : IsSymmetricRel (Set.univ : Set (α × α)) := by
simp [IsSymmetricRel] | lemma | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | isSymmetricRel_univ | null |
IsSymmetricRel.preimage_prodMap {U : Set (β × β)} (ht : IsSymmetricRel U) (f : α → β) :
IsSymmetricRel (Prod.map f f ⁻¹' U) :=
Set.ext fun _ ↦ ht.mk_mem_comm | lemma | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | IsSymmetricRel.preimage_prodMap | null |
IsSymmetricRel.image_prodMap {U : Set (α × α)} (ht : IsSymmetricRel U) (f : α → β) :
IsSymmetricRel (Prod.map f f '' U) := by
rw [IsSymmetricRel, ← image_swap_eq_preimage_swap, ← image_comp, ← Prod.map_comp_swap, image_comp,
image_swap_eq_preimage_swap, ht] | lemma | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | IsSymmetricRel.image_prodMap | null |
IsSymmetricRel.prod_subset_comm {s : Set (α × α)} {t u : Set α} (hs : IsSymmetricRel s) :
t ×ˢ u ⊆ s ↔ u ×ˢ t ⊆ s := by
rw [← hs.eq, ← image_subset_iff, image_swap_prod, hs.eq] | lemma | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | IsSymmetricRel.prod_subset_comm | null |
IsSymmetricRel.mem_filter_prod_comm {s : Set (α × α)} {f g : Filter α}
(hs : IsSymmetricRel s) :
s ∈ f ×ˢ g ↔ s ∈ g ×ˢ f := by
rw [← hs.eq, ← mem_map, ← prod_comm, hs.eq] | lemma | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | IsSymmetricRel.mem_filter_prod_comm | null |
UniformSpace.Core (α : Type u) where
/-- The uniformity filter. Once `UniformSpace` is defined, `𝓤 α` (`_root_.uniformity`) becomes the
normal form. -/
uniformity : Filter (α × α)
/-- Every set in the uniformity filter includes the diagonal. -/
refl : 𝓟 idRel ≤ uniformity
/-- If `s ∈ uniformity`, then `Prod.swap ⁻¹' s ∈ uniformity`. -/
symm : Tendsto Prod.swap uniformity uniformity
/-- For every set `u ∈ uniformity`, there exists `v ∈ uniformity` such that `v ○ v ⊆ u`. -/
comp : (uniformity.lift' fun s => s ○ s) ≤ uniformity | structure | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.Core | This core description of a uniform space is outside of the type class hierarchy. It is useful
for constructions of uniform spaces, when the topology is derived from the uniform space. |
protected UniformSpace.Core.comp_mem_uniformity_sets {c : Core α} {s : Set (α × α)}
(hs : s ∈ c.uniformity) : ∃ t ∈ c.uniformity, t ○ t ⊆ s :=
(mem_lift'_sets <| monotone_id.compRel monotone_id).mp <| c.comp hs | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.Core.comp_mem_uniformity_sets | null |
UniformSpace.Core.mk' {α : Type u} (U : Filter (α × α)) (refl : ∀ r ∈ U, ∀ (x), (x, x) ∈ r)
(symm : ∀ r ∈ U, Prod.swap ⁻¹' r ∈ U) (comp : ∀ r ∈ U, ∃ t ∈ U, t ○ t ⊆ r) :
UniformSpace.Core α :=
⟨U, fun _r ru => idRel_subset.2 (refl _ ru), symm, fun _r ru =>
let ⟨_s, hs, hsr⟩ := comp _ ru
mem_of_superset (mem_lift' hs) hsr⟩ | def | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.Core.mk' | An alternative constructor for `UniformSpace.Core`. This version unfolds various
`Filter`-related definitions. |
UniformSpace.Core.mkOfBasis {α : Type u} (B : FilterBasis (α × α))
(refl : ∀ r ∈ B, ∀ (x), (x, x) ∈ r) (symm : ∀ r ∈ B, ∃ t ∈ B, t ⊆ Prod.swap ⁻¹' r)
(comp : ∀ r ∈ B, ∃ t ∈ B, t ○ t ⊆ r) : UniformSpace.Core α where
uniformity := B.filter
refl := B.hasBasis.ge_iff.mpr fun _r ru => idRel_subset.2 <| refl _ ru
symm := (B.hasBasis.tendsto_iff B.hasBasis).mpr symm
comp := (HasBasis.le_basis_iff (B.hasBasis.lift' (monotone_id.compRel monotone_id))
B.hasBasis).2 comp | def | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.Core.mkOfBasis | Defining a `UniformSpace.Core` from a filter basis satisfying some uniformity-like axioms. |
UniformSpace.Core.toTopologicalSpace {α : Type u} (u : UniformSpace.Core α) :
TopologicalSpace α :=
.mkOfNhds fun x ↦ .comap (Prod.mk x) u.uniformity | def | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.Core.toTopologicalSpace | A uniform space generates a topological space |
UniformSpace.Core.ext :
∀ {u₁ u₂ : UniformSpace.Core α}, u₁.uniformity = u₂.uniformity → u₁ = u₂
| ⟨_, _, _, _⟩, ⟨_, _, _, _⟩, rfl => rfl | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.Core.ext | null |
UniformSpace.Core.nhds_toTopologicalSpace {α : Type u} (u : Core α) (x : α) :
@nhds α u.toTopologicalSpace x = comap (Prod.mk x) u.uniformity := by
apply TopologicalSpace.nhds_mkOfNhds_of_hasBasis (fun _ ↦ (basis_sets _).comap _)
· exact fun a U hU ↦ u.refl hU rfl
· intro a U hU
rcases u.comp_mem_uniformity_sets hU with ⟨V, hV, hVU⟩
filter_upwards [preimage_mem_comap hV] with b hb
filter_upwards [preimage_mem_comap hV] with c hc
exact hVU ⟨b, hb, hc⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.Core.nhds_toTopologicalSpace | null |
UniformSpace (α : Type u) extends TopologicalSpace α where
/-- The uniformity filter. -/
protected uniformity : Filter (α × α)
/-- If `s ∈ uniformity`, then `Prod.swap ⁻¹' s ∈ uniformity`. -/
protected symm : Tendsto Prod.swap uniformity uniformity
/-- For every set `u ∈ uniformity`, there exists `v ∈ uniformity` such that `v ○ v ⊆ u`. -/
protected comp : (uniformity.lift' fun s => s ○ s) ≤ uniformity
/-- The uniformity agrees with the topology: the neighborhoods filter of each point `x`
is equal to `Filter.comap (Prod.mk x) (𝓤 α)`. -/
protected nhds_eq_comap_uniformity (x : α) : 𝓝 x = comap (Prod.mk x) uniformity | class | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace | A uniform space is a generalization of the "uniform" topological aspects of a
metric space. It consists of a filter on `α × α` called the "uniformity", which
satisfies properties analogous to the reflexivity, symmetry, and triangle properties
of a metric.
A metric space has a natural uniformity, and a uniform space has a natural topology.
A topological group also has a natural uniformity, even when it is not metrizable. |
uniformity (α : Type u) [UniformSpace α] : Filter (α × α) :=
@UniformSpace.uniformity α _ | def | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | uniformity | The uniformity is a filter on α × α (inferred from an ambient uniform space
structure on α). |
UniformSpace.ofCoreEq {α : Type u} (u : UniformSpace.Core α) (t : TopologicalSpace α)
(h : t = u.toTopologicalSpace) : UniformSpace α where
__ := u
toTopologicalSpace := t
nhds_eq_comap_uniformity x := by rw [h, u.nhds_toTopologicalSpace] | abbrev | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.ofCoreEq | Notation for the uniformity filter with respect to a non-standard `UniformSpace` instance. -/
scoped[Uniformity] notation "𝓤[" u "]" => @uniformity _ u
@[inherit_doc]
scoped[Uniformity] notation "𝓤" => uniformity
/-- Construct a `UniformSpace` from a `u : UniformSpace.Core` and a `TopologicalSpace` structure
that is equal to `u.toTopologicalSpace`. |
UniformSpace.ofCore {α : Type u} (u : UniformSpace.Core α) : UniformSpace α :=
.ofCoreEq u _ rfl | abbrev | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.ofCore | Construct a `UniformSpace` from a `UniformSpace.Core`. |
UniformSpace.toCore (u : UniformSpace α) : UniformSpace.Core α where
__ := u
refl := by
rintro U hU ⟨x, y⟩ (rfl : x = y)
have : Prod.mk x ⁻¹' U ∈ 𝓝 x := by
rw [UniformSpace.nhds_eq_comap_uniformity]
exact preimage_mem_comap hU
convert mem_of_mem_nhds this | abbrev | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.toCore | Construct a `UniformSpace.Core` from a `UniformSpace`. |
UniformSpace.toCore_toTopologicalSpace (u : UniformSpace α) :
u.toCore.toTopologicalSpace = u.toTopologicalSpace :=
TopologicalSpace.ext_nhds fun a ↦ by
rw [u.nhds_eq_comap_uniformity, u.toCore.nhds_toTopologicalSpace] | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.toCore_toTopologicalSpace | null |
UniformSpace.mem_uniformity_ofCore_iff {u : UniformSpace.Core α} {s : Set (α × α)} :
s ∈ 𝓤[.ofCore u] ↔ s ∈ u.uniformity :=
Iff.rfl
@[ext (iff := false)] | lemma | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.mem_uniformity_ofCore_iff | null |
protected UniformSpace.ext {u₁ u₂ : UniformSpace α} (h : 𝓤[u₁] = 𝓤[u₂]) : u₁ = u₂ := by
have : u₁.toTopologicalSpace = u₂.toTopologicalSpace := TopologicalSpace.ext_nhds fun x ↦ by
rw [u₁.nhds_eq_comap_uniformity, u₂.nhds_eq_comap_uniformity]
exact congr_arg (comap _) h
cases u₁; cases u₂; congr | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.ext | null |
protected UniformSpace.ext_iff {u₁ u₂ : UniformSpace α} :
u₁ = u₂ ↔ ∀ s, s ∈ 𝓤[u₁] ↔ s ∈ 𝓤[u₂] :=
⟨fun h _ => h ▸ Iff.rfl, fun h => by ext; exact h _⟩ | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.ext_iff | null |
UniformSpace.ofCoreEq_toCore (u : UniformSpace α) (t : TopologicalSpace α)
(h : t = u.toCore.toTopologicalSpace) : .ofCoreEq u.toCore t h = u :=
UniformSpace.ext rfl | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.ofCoreEq_toCore | null |
UniformSpace.replaceTopology {α : Type*} [i : TopologicalSpace α] (u : UniformSpace α)
(h : i = u.toTopologicalSpace) : UniformSpace α where
__ := u
toTopologicalSpace := i
nhds_eq_comap_uniformity x := by rw [h, u.nhds_eq_comap_uniformity] | abbrev | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.replaceTopology | Replace topology in a `UniformSpace` instance with a propositionally (but possibly not
definitionally) equal one. |
UniformSpace.replaceTopology_eq {α : Type*} [i : TopologicalSpace α] (u : UniformSpace α)
(h : i = u.toTopologicalSpace) : u.replaceTopology h = u :=
UniformSpace.ext rfl | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | UniformSpace.replaceTopology_eq | null |
nhds_eq_comap_uniformity {x : α} : 𝓝 x = (𝓤 α).comap (Prod.mk x) :=
UniformSpace.nhds_eq_comap_uniformity x | 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 |
isOpen_uniformity {s : Set α} :
IsOpen s ↔ ∀ x ∈ s, { p : α × α | p.1 = x → p.2 ∈ s } ∈ 𝓤 α := by
simp only [isOpen_iff_mem_nhds, 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 | isOpen_uniformity | null |
refl_le_uniformity : 𝓟 idRel ≤ 𝓤 α :=
(@UniformSpace.toCore α _).refl | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | refl_le_uniformity | null |
uniformity.neBot [Nonempty α] : NeBot (𝓤 α) :=
diagonal_nonempty.principal_neBot.mono refl_le_uniformity | instance | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | uniformity.neBot | null |
refl_mem_uniformity {x : α} {s : Set (α × α)} (h : s ∈ 𝓤 α) : (x, x) ∈ s :=
refl_le_uniformity h rfl | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | refl_mem_uniformity | null |
mem_uniformity_of_eq {x y : α} {s : Set (α × α)} (h : s ∈ 𝓤 α) (hx : x = y) : (x, y) ∈ s :=
refl_le_uniformity h hx | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | mem_uniformity_of_eq | null |
symm_le_uniformity : map (@Prod.swap α α) (𝓤 _) ≤ 𝓤 _ :=
UniformSpace.symm | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | symm_le_uniformity | null |
comp_le_uniformity : ((𝓤 α).lift' fun s : Set (α × α) => s ○ s) ≤ 𝓤 α :=
UniformSpace.comp | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | comp_le_uniformity | null |
lift'_comp_uniformity : ((𝓤 α).lift' fun s : Set (α × α) => s ○ s) = 𝓤 α :=
comp_le_uniformity.antisymm <| le_lift'.2 fun _s hs ↦ mem_of_superset hs <|
subset_comp_self <| idRel_subset.2 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 | lift'_comp_uniformity | null |
tendsto_swap_uniformity : Tendsto (@Prod.swap α α) (𝓤 α) (𝓤 α) :=
symm_le_uniformity | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | tendsto_swap_uniformity | null |
comp_mem_uniformity_sets {s : Set (α × α)} (hs : s ∈ 𝓤 α) : ∃ t ∈ 𝓤 α, t ○ t ⊆ s :=
(mem_lift'_sets <| monotone_id.compRel monotone_id).mp <| comp_le_uniformity hs | theorem | Topology | [
"Mathlib.Algebra.Group.Defs",
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Topology.Order"
] | Mathlib/Topology/UniformSpace/Defs.lean | comp_mem_uniformity_sets | null |
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