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 ⌀ |
|---|---|---|---|---|---|---|
toTopologicalSpace_inf {u v : UniformSpace α} :
(u ⊓ v).toTopologicalSpace = u.toTopologicalSpace ⊓ v.toTopologicalSpace :=
rfl | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | toTopologicalSpace_inf | null |
UniformContinuous.continuous [UniformSpace α] [UniformSpace β] {f : α → β}
(hf : UniformContinuous f) : Continuous f :=
continuous_iff_le_induced.mpr <| UniformSpace.toTopologicalSpace_mono <|
uniformContinuous_iff.1 hf | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | UniformContinuous.continuous | null |
ULift.uniformSpace [UniformSpace α] : UniformSpace (ULift α) :=
UniformSpace.comap ULift.down ‹_› | instance | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | ULift.uniformSpace | Uniform space structure on `ULift α`. |
OrderDual.instUniformSpace [UniformSpace α] : UniformSpace (αᵒᵈ) :=
‹UniformSpace α› | instance | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | OrderDual.instUniformSpace | Uniform space structure on `αᵒᵈ`. |
UniformContinuous.inf_rng {f : α → β} {u₁ : UniformSpace α} {u₂ u₃ : UniformSpace β}
(h₁ : UniformContinuous[u₁, u₂] f) (h₂ : UniformContinuous[u₁, u₃] f) :
UniformContinuous[u₁, u₂ ⊓ u₃] f :=
tendsto_inf.mpr ⟨h₁, h₂⟩ | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | UniformContinuous.inf_rng | null |
UniformContinuous.inf_dom_left {f : α → β} {u₁ u₂ : UniformSpace α} {u₃ : UniformSpace β}
(hf : UniformContinuous[u₁, u₃] f) : UniformContinuous[u₁ ⊓ u₂, u₃] f :=
tendsto_inf_left hf | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | UniformContinuous.inf_dom_left | null |
UniformContinuous.inf_dom_right {f : α → β} {u₁ u₂ : UniformSpace α} {u₃ : UniformSpace β}
(hf : UniformContinuous[u₂, u₃] f) : UniformContinuous[u₁ ⊓ u₂, u₃] f :=
tendsto_inf_right hf | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | UniformContinuous.inf_dom_right | null |
uniformContinuous_sInf_dom {f : α → β} {u₁ : Set (UniformSpace α)} {u₂ : UniformSpace β}
{u : UniformSpace α} (h₁ : u ∈ u₁) (hf : UniformContinuous[u, u₂] f) :
UniformContinuous[sInf u₁, u₂] f := by
delta UniformContinuous
rw [sInf_eq_iInf', iInf_uniformity]
exact tendsto_iInf' ⟨u, h₁⟩ hf | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous_sInf_dom | null |
uniformContinuous_sInf_rng {f : α → β} {u₁ : UniformSpace α} {u₂ : Set (UniformSpace β)} :
UniformContinuous[u₁, sInf u₂] f ↔ ∀ u ∈ u₂, UniformContinuous[u₁, u] f := by
delta UniformContinuous
rw [sInf_eq_iInf', iInf_uniformity, tendsto_iInf, SetCoe.forall] | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous_sInf_rng | null |
uniformContinuous_iInf_dom {f : α → β} {u₁ : ι → UniformSpace α} {u₂ : UniformSpace β}
{i : ι} (hf : UniformContinuous[u₁ i, u₂] f) : UniformContinuous[iInf u₁, u₂] f := by
delta UniformContinuous
rw [iInf_uniformity]
exact tendsto_iInf' i hf | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous_iInf_dom | null |
uniformContinuous_iInf_rng {f : α → β} {u₁ : UniformSpace α} {u₂ : ι → UniformSpace β} :
UniformContinuous[u₁, iInf u₂] f ↔ ∀ i, UniformContinuous[u₁, u₂ i] f := by
delta UniformContinuous
rw [iInf_uniformity, tendsto_iInf] | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous_iInf_rng | null |
discreteTopology_of_discrete_uniformity [hα : UniformSpace α] (h : uniformity α = 𝓟 idRel) :
DiscreteTopology α :=
⟨(UniformSpace.ext h.symm : ⊥ = hα) ▸ rfl⟩ | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | discreteTopology_of_discrete_uniformity | A uniform space with the discrete uniformity has the discrete topology. |
uniformContinuous_ofMul : UniformContinuous (ofMul : α → Additive α) :=
uniformContinuous_id | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous_ofMul | null |
uniformContinuous_toMul : UniformContinuous (toMul : Additive α → α) :=
uniformContinuous_id | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous_toMul | null |
uniformContinuous_ofAdd : UniformContinuous (ofAdd : α → Multiplicative α) :=
uniformContinuous_id | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous_ofAdd | null |
uniformContinuous_toAdd : UniformContinuous (toAdd : Multiplicative α → α) :=
uniformContinuous_id | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous_toAdd | null |
uniformity_additive : 𝓤 (Additive α) = (𝓤 α).map (Prod.map ofMul ofMul) := rfl | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformity_additive | null |
uniformity_multiplicative : 𝓤 (Multiplicative α) = (𝓤 α).map (Prod.map ofAdd ofAdd) := rfl | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformity_multiplicative | null |
instUniformSpaceSubtype {p : α → Prop} [t : UniformSpace α] : UniformSpace (Subtype p) :=
UniformSpace.comap Subtype.val t | instance | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | instUniformSpaceSubtype | null |
uniformity_subtype {p : α → Prop} [UniformSpace α] :
𝓤 (Subtype p) = comap (fun q : Subtype p × Subtype p => (q.1.1, q.2.1)) (𝓤 α) :=
rfl | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformity_subtype | null |
uniformity_setCoe {s : Set α} [UniformSpace α] :
𝓤 s = comap (Prod.map ((↑) : s → α) ((↑) : s → α)) (𝓤 α) :=
rfl | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformity_setCoe | null |
map_uniformity_set_coe {s : Set α} [UniformSpace α] :
map (Prod.map (↑) (↑)) (𝓤 s) = 𝓤 α ⊓ 𝓟 (s ×ˢ s) := by
rw [uniformity_setCoe, map_comap, range_prodMap, Subtype.range_val] | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | map_uniformity_set_coe | null |
uniformContinuous_subtype_val {p : α → Prop} [UniformSpace α] :
UniformContinuous (Subtype.val : { a : α // p a } → α) :=
uniformContinuous_comap | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous_subtype_val | null |
UniformContinuous.subtype_mk {p : α → Prop} [UniformSpace α] [UniformSpace β] {f : β → α}
(hf : UniformContinuous f) (h : ∀ x, p (f x)) :
UniformContinuous (fun x => ⟨f x, h x⟩ : β → Subtype p) :=
uniformContinuous_comap' hf | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | UniformContinuous.subtype_mk | null |
uniformContinuousOn_iff_restrict [UniformSpace α] [UniformSpace β] {f : α → β} {s : Set α} :
UniformContinuousOn f s ↔ UniformContinuous (s.restrict f) := by
delta UniformContinuousOn UniformContinuous
rw [← map_uniformity_set_coe, tendsto_map'_iff]; rfl | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuousOn_iff_restrict | null |
tendsto_of_uniformContinuous_subtype [UniformSpace α] [UniformSpace β] {f : α → β}
{s : Set α} {a : α} (hf : UniformContinuous fun x : s => f x.val) (ha : s ∈ 𝓝 a) :
Tendsto f (𝓝 a) (𝓝 (f a)) := by
rw [(@map_nhds_subtype_coe_eq_nhds α _ s a (mem_of_mem_nhds ha) ha).symm]
exact tendsto_map' hf.continuous.continuousAt | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | tendsto_of_uniformContinuous_subtype | null |
UniformContinuousOn.continuousOn [UniformSpace α] [UniformSpace β] {f : α → β} {s : Set α}
(h : UniformContinuousOn f s) : ContinuousOn f s := by
rw [uniformContinuousOn_iff_restrict] at h
rw [continuousOn_iff_continuous_restrict]
exact h.continuous
@[to_additive] | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | UniformContinuousOn.continuousOn | null |
@[to_additive]
uniformity_mulOpposite [UniformSpace α] :
𝓤 αᵐᵒᵖ = comap (fun q : αᵐᵒᵖ × αᵐᵒᵖ => (q.1.unop, q.2.unop)) (𝓤 α) :=
rfl
@[to_additive (attr := simp)] | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformity_mulOpposite | null |
comap_uniformity_mulOpposite [UniformSpace α] :
comap (fun p : α × α => (MulOpposite.op p.1, MulOpposite.op p.2)) (𝓤 αᵐᵒᵖ) = 𝓤 α := by
simpa [uniformity_mulOpposite, comap_comap, (· ∘ ·)] using comap_id | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | comap_uniformity_mulOpposite | null |
@[to_additive]
uniformContinuous_unop [UniformSpace α] : UniformContinuous (unop : αᵐᵒᵖ → α) :=
uniformContinuous_comap
@[to_additive] | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous_unop | null |
uniformContinuous_op [UniformSpace α] : UniformContinuous (op : α → αᵐᵒᵖ) :=
uniformContinuous_comap' uniformContinuous_id | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous_op | null |
instUniformSpaceProd [u₁ : UniformSpace α] [u₂ : UniformSpace β] : UniformSpace (α × β) :=
u₁.comap Prod.fst ⊓ u₂.comap Prod.snd | instance | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | instUniformSpaceProd | null |
uniformity_prod [UniformSpace α] [UniformSpace β] :
𝓤 (α × β) =
((𝓤 α).comap fun p : (α × β) × α × β => (p.1.1, p.2.1)) ⊓
(𝓤 β).comap fun p : (α × β) × α × β => (p.1.2, p.2.2) :=
rfl | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformity_prod | null |
uniformity_prod_eq_comap_prod [UniformSpace α] [UniformSpace β] :
𝓤 (α × β) =
comap (fun p : (α × β) × α × β => ((p.1.1, p.2.1), (p.1.2, p.2.2))) (𝓤 α ×ˢ 𝓤 β) := by
simp_rw [uniformity_prod, prod_eq_inf, Filter.comap_inf, Filter.comap_comap, Function.comp_def] | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformity_prod_eq_comap_prod | null |
uniformity_prod_eq_prod [UniformSpace α] [UniformSpace β] :
𝓤 (α × β) = map (fun p : (α × α) × β × β => ((p.1.1, p.2.1), (p.1.2, p.2.2))) (𝓤 α ×ˢ 𝓤 β) := by
rw [map_swap4_eq_comap, uniformity_prod_eq_comap_prod] | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformity_prod_eq_prod | null |
mem_uniformity_of_uniformContinuous_invariant [UniformSpace α] [UniformSpace β]
{s : Set (β × β)} {f : α → α → β} (hf : UniformContinuous fun p : α × α => f p.1 p.2)
(hs : s ∈ 𝓤 β) : ∃ u ∈ 𝓤 α, ∀ a b c, (a, b) ∈ u → (f a c, f b c) ∈ s := by
rw [UniformContinuous, uniformity_prod_eq_prod, tendsto_map'_iff] at hf
rcases mem_prod_iff.1 (mem_map.1 <| hf hs) with ⟨u, hu, v, hv, huvt⟩
exact ⟨u, hu, fun a b c hab => @huvt ((_, _), (_, _)) ⟨hab, refl_mem_uniformity hv⟩⟩ | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | mem_uniformity_of_uniformContinuous_invariant | null |
entourageProd (u : Set (α × α)) (v : Set (β × β)) : Set ((α × β) × α × β) :=
{((a₁, b₁),(a₂, b₂)) | (a₁, a₂) ∈ u ∧ (b₁, b₂) ∈ v} | def | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | entourageProd | An entourage of the diagonal in `α` and an entourage in `β` yield an entourage in `α × β`
once we permute coordinates. |
mem_entourageProd {u : Set (α × α)} {v : Set (β × β)} {p : (α × β) × α × β} :
p ∈ entourageProd u v ↔ (p.1.1, p.2.1) ∈ u ∧ (p.1.2, p.2.2) ∈ v := Iff.rfl | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | mem_entourageProd | null |
entourageProd_mem_uniformity [t₁ : UniformSpace α] [t₂ : UniformSpace β] {u : Set (α × α)}
{v : Set (β × β)} (hu : u ∈ 𝓤 α) (hv : v ∈ 𝓤 β) :
entourageProd u v ∈ 𝓤 (α × β) := by
rw [uniformity_prod]; exact inter_mem_inf (preimage_mem_comap hu) (preimage_mem_comap hv) | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | entourageProd_mem_uniformity | null |
ball_entourageProd (u : Set (α × α)) (v : Set (β × β)) (x : α × β) :
ball x (entourageProd u v) = ball x.1 u ×ˢ ball x.2 v := by
ext p; simp only [ball, entourageProd, Set.mem_setOf_eq, Set.mem_prod, Set.mem_preimage] | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | ball_entourageProd | null |
IsSymmetricRel.entourageProd {u : Set (α × α)} {v : Set (β × β)}
(hu : IsSymmetricRel u) (hv : IsSymmetricRel v) :
IsSymmetricRel (entourageProd u v) :=
Set.ext fun _ ↦ and_congr hu.mk_mem_comm hv.mk_mem_comm | lemma | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | IsSymmetricRel.entourageProd | null |
Filter.HasBasis.uniformity_prod {ιa ιb : Type*} [UniformSpace α] [UniformSpace β]
{pa : ιa → Prop} {pb : ιb → Prop} {sa : ιa → Set (α × α)} {sb : ιb → Set (β × β)}
(ha : (𝓤 α).HasBasis pa sa) (hb : (𝓤 β).HasBasis pb sb) :
(𝓤 (α × β)).HasBasis (fun i : ιa × ιb ↦ pa i.1 ∧ pb i.2)
(fun i ↦ entourageProd (sa i.1) (sb i.2)) :=
(ha.comap _).inf (hb.comap _) | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | Filter.HasBasis.uniformity_prod | null |
entourageProd_subset [UniformSpace α] [UniformSpace β]
{s : Set ((α × β) × α × β)} (h : s ∈ 𝓤 (α × β)) :
∃ u ∈ 𝓤 α, ∃ v ∈ 𝓤 β, entourageProd u v ⊆ s := by
rcases (((𝓤 α).basis_sets.uniformity_prod (𝓤 β).basis_sets).mem_iff' s).1 h with ⟨w, hw⟩
use w.1, hw.1.1, w.2, hw.1.2, hw.2 | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | entourageProd_subset | null |
tendsto_prod_uniformity_fst [UniformSpace α] [UniformSpace β] :
Tendsto (fun p : (α × β) × α × β => (p.1.1, p.2.1)) (𝓤 (α × β)) (𝓤 α) :=
le_trans (map_mono inf_le_left) map_comap_le | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | tendsto_prod_uniformity_fst | null |
tendsto_prod_uniformity_snd [UniformSpace α] [UniformSpace β] :
Tendsto (fun p : (α × β) × α × β => (p.1.2, p.2.2)) (𝓤 (α × β)) (𝓤 β) :=
le_trans (map_mono inf_le_right) map_comap_le | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | tendsto_prod_uniformity_snd | null |
uniformContinuous_fst [UniformSpace α] [UniformSpace β] :
UniformContinuous fun p : α × β => p.1 :=
tendsto_prod_uniformity_fst | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous_fst | null |
uniformContinuous_snd [UniformSpace α] [UniformSpace β] :
UniformContinuous fun p : α × β => p.2 :=
tendsto_prod_uniformity_snd
variable [UniformSpace α] [UniformSpace β] [UniformSpace γ] | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous_snd | null |
UniformContinuous.prodMk {f₁ : α → β} {f₂ : α → γ} (h₁ : UniformContinuous f₁)
(h₂ : UniformContinuous f₂) : UniformContinuous fun a => (f₁ a, f₂ a) := by
rw [UniformContinuous, uniformity_prod]
exact tendsto_inf.2 ⟨tendsto_comap_iff.2 h₁, tendsto_comap_iff.2 h₂⟩
@[deprecated (since := "2025-03-10")]
alias UniformContinuous.prod_mk := UniformContinuous.prodMk | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | UniformContinuous.prodMk | null |
UniformContinuous.prodMk_left {f : α × β → γ} (h : UniformContinuous f) (b) :
UniformContinuous fun a => f (a, b) :=
h.comp (uniformContinuous_id.prodMk uniformContinuous_const)
@[deprecated (since := "2025-03-10")]
alias UniformContinuous.prod_mk_left := UniformContinuous.prodMk_left | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | UniformContinuous.prodMk_left | null |
UniformContinuous.prodMk_right {f : α × β → γ} (h : UniformContinuous f) (a) :
UniformContinuous fun b => f (a, b) :=
h.comp (uniformContinuous_const.prodMk uniformContinuous_id)
@[deprecated (since := "2025-03-10")]
alias UniformContinuous.prod_mk_right := UniformContinuous.prodMk_right | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | UniformContinuous.prodMk_right | null |
UniformContinuous.prodMap [UniformSpace δ] {f : α → γ} {g : β → δ}
(hf : UniformContinuous f) (hg : UniformContinuous g) : UniformContinuous (Prod.map f g) :=
(hf.comp uniformContinuous_fst).prodMk (hg.comp uniformContinuous_snd) | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | UniformContinuous.prodMap | null |
toTopologicalSpace_prod {α} {β} [u : UniformSpace α] [v : UniformSpace β] :
@UniformSpace.toTopologicalSpace (α × β) instUniformSpaceProd =
@instTopologicalSpaceProd α β u.toTopologicalSpace v.toTopologicalSpace :=
rfl | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | toTopologicalSpace_prod | null |
uniformContinuous_inf_dom_left₂ {α β γ} {f : α → β → γ} {ua1 ua2 : UniformSpace α}
{ub1 ub2 : UniformSpace β} {uc1 : UniformSpace γ}
(h : by haveI := ua1; haveI := ub1; exact UniformContinuous fun p : α × β => f p.1 p.2) : by
haveI := ua1 ⊓ ua2; haveI := ub1 ⊓ ub2
exact UniformContinuous fun p : α × β => f p.1 p.2 := by
have ha := @UniformContinuous.inf_dom_left _ _ id ua1 ua2 ua1 (@uniformContinuous_id _ (id _))
have hb := @UniformContinuous.inf_dom_left _ _ id ub1 ub2 ub1 (@uniformContinuous_id _ (id _))
have h_unif_cont_id :=
@UniformContinuous.prodMap _ _ _ _ (ua1 ⊓ ua2) (ub1 ⊓ ub2) ua1 ub1 _ _ ha hb
exact @UniformContinuous.comp _ _ _ (id _) (id _) _ _ _ h h_unif_cont_id | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous_inf_dom_left₂ | A version of `UniformContinuous.inf_dom_left` for binary functions |
uniformContinuous_inf_dom_right₂ {α β γ} {f : α → β → γ} {ua1 ua2 : UniformSpace α}
{ub1 ub2 : UniformSpace β} {uc1 : UniformSpace γ}
(h : by haveI := ua2; haveI := ub2; exact UniformContinuous fun p : α × β => f p.1 p.2) : by
haveI := ua1 ⊓ ua2; haveI := ub1 ⊓ ub2
exact UniformContinuous fun p : α × β => f p.1 p.2 := by
have ha := @UniformContinuous.inf_dom_right _ _ id ua1 ua2 ua2 (@uniformContinuous_id _ (id _))
have hb := @UniformContinuous.inf_dom_right _ _ id ub1 ub2 ub2 (@uniformContinuous_id _ (id _))
have h_unif_cont_id :=
@UniformContinuous.prodMap _ _ _ _ (ua1 ⊓ ua2) (ub1 ⊓ ub2) ua2 ub2 _ _ ha hb
exact @UniformContinuous.comp _ _ _ (id _) (id _) _ _ _ h h_unif_cont_id | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous_inf_dom_right₂ | A version of `UniformContinuous.inf_dom_right` for binary functions |
uniformContinuous_sInf_dom₂ {α β γ} {f : α → β → γ} {uas : Set (UniformSpace α)}
{ubs : Set (UniformSpace β)} {ua : UniformSpace α} {ub : UniformSpace β} {uc : UniformSpace γ}
(ha : ua ∈ uas) (hb : ub ∈ ubs) (hf : UniformContinuous fun p : α × β => f p.1 p.2) : by
haveI := sInf uas; haveI := sInf ubs
exact @UniformContinuous _ _ _ uc fun p : α × β => f p.1 p.2 := by
let _ : UniformSpace (α × β) := instUniformSpaceProd
have ha := uniformContinuous_sInf_dom ha uniformContinuous_id
have hb := uniformContinuous_sInf_dom hb uniformContinuous_id
have h_unif_cont_id := @UniformContinuous.prodMap _ _ _ _ (sInf uas) (sInf ubs) ua ub _ _ ha hb
exact @UniformContinuous.comp _ _ _ (id _) (id _) _ _ _ hf h_unif_cont_id | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous_sInf_dom₂ | A version of `uniformContinuous_sInf_dom` for binary functions |
UniformContinuous₂ (f : α → β → γ) :=
UniformContinuous (uncurry f) | def | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | UniformContinuous₂ | Uniform continuity for functions of two variables. |
uniformContinuous₂_def (f : α → β → γ) :
UniformContinuous₂ f ↔ UniformContinuous (uncurry f) :=
Iff.rfl | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous₂_def | null |
UniformContinuous₂.uniformContinuous {f : α → β → γ} (h : UniformContinuous₂ f) :
UniformContinuous (uncurry f) :=
h | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | UniformContinuous₂.uniformContinuous | null |
uniformContinuous₂_curry (f : α × β → γ) :
UniformContinuous₂ (Function.curry f) ↔ UniformContinuous f := by
rw [UniformContinuous₂, uncurry_curry] | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous₂_curry | null |
UniformContinuous₂.comp {f : α → β → γ} {g : γ → δ} (hg : UniformContinuous g)
(hf : UniformContinuous₂ f) : UniformContinuous₂ (g ∘₂ f) :=
hg.comp hf | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | UniformContinuous₂.comp | null |
UniformContinuous₂.bicompl {f : α → β → γ} {ga : δ → α} {gb : δ' → β}
(hf : UniformContinuous₂ f) (hga : UniformContinuous ga) (hgb : UniformContinuous gb) :
UniformContinuous₂ (bicompl f ga gb) :=
hf.uniformContinuous.comp (hga.prodMap hgb) | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | UniformContinuous₂.bicompl | null |
toTopologicalSpace_subtype [u : UniformSpace α] {p : α → Prop} :
@UniformSpace.toTopologicalSpace (Subtype p) instUniformSpaceSubtype =
@instTopologicalSpaceSubtype α p u.toTopologicalSpace :=
rfl | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | toTopologicalSpace_subtype | null |
Sum.instUniformSpace : UniformSpace (α ⊕ β) where
uniformity := map (fun p : α × α => (inl p.1, inl p.2)) (𝓤 α) ⊔
map (fun p : β × β => (inr p.1, inr p.2)) (𝓤 β)
symm := fun _ hs ↦ ⟨symm_le_uniformity hs.1, symm_le_uniformity hs.2⟩
comp := fun s hs ↦ by
rcases comp_mem_uniformity_sets hs.1 with ⟨tα, htα, Htα⟩
rcases comp_mem_uniformity_sets hs.2 with ⟨tβ, htβ, Htβ⟩
filter_upwards [mem_lift' (union_mem_sup (image_mem_map htα) (image_mem_map htβ))]
rintro ⟨_, _⟩ ⟨z, ⟨⟨a, b⟩, hab, ⟨⟩⟩ | ⟨⟨a, b⟩, hab, ⟨⟩⟩, ⟨⟨_, c⟩, hbc, ⟨⟩⟩ | ⟨⟨_, c⟩, hbc, ⟨⟩⟩⟩
exacts [@Htα (_, _) ⟨b, hab, hbc⟩, @Htβ (_, _) ⟨b, hab, hbc⟩]
nhds_eq_comap_uniformity x := by
ext
cases x <;> simp [mem_comap', -mem_comap, nhds_inl, nhds_inr, nhds_eq_comap_uniformity,
Prod.ext_iff] | instance | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | Sum.instUniformSpace | Uniformity on a disjoint union. Entourages of the diagonal in the union are obtained
by taking independently an entourage of the diagonal in the first part, and an entourage of
the diagonal in the second part. |
union_mem_uniformity_sum {a : Set (α × α)} (ha : a ∈ 𝓤 α) {b : Set (β × β)} (hb : b ∈ 𝓤 β) :
Prod.map inl inl '' a ∪ Prod.map inr inr '' b ∈ 𝓤 (α ⊕ β) :=
union_mem_sup (image_mem_map ha) (image_mem_map hb) | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | union_mem_uniformity_sum | The union of an entourage of the diagonal in each set of a disjoint union is again an entourage
of the diagonal. |
Sum.uniformity : 𝓤 (α ⊕ β) = map (Prod.map inl inl) (𝓤 α) ⊔ map (Prod.map inr inr) (𝓤 β) :=
rfl | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | Sum.uniformity | null |
uniformContinuous_inl : UniformContinuous (Sum.inl : α → α ⊕ β) := le_sup_left | lemma | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous_inl | null |
uniformContinuous_inr : UniformContinuous (Sum.inr : β → α ⊕ β) := le_sup_right | lemma | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | uniformContinuous_inr | null |
tendsto_nhds_right {f : Filter β} {u : β → α} {a : α} :
Tendsto u f (𝓝 a) ↔ Tendsto (fun x => (a, u x)) f (𝓤 α) := by
rw [nhds_eq_comap_uniformity, tendsto_comap_iff]; rfl | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | tendsto_nhds_right | null |
tendsto_nhds_left {f : Filter β} {u : β → α} {a : α} :
Tendsto u f (𝓝 a) ↔ Tendsto (fun x => (u x, a)) f (𝓤 α) := by
rw [nhds_eq_comap_uniformity', tendsto_comap_iff]; rfl | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | tendsto_nhds_left | null |
continuousAt_iff'_right [TopologicalSpace β] {f : β → α} {b : β} :
ContinuousAt f b ↔ Tendsto (fun x => (f b, f x)) (𝓝 b) (𝓤 α) := by
rw [ContinuousAt, tendsto_nhds_right] | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | continuousAt_iff'_right | null |
continuousAt_iff'_left [TopologicalSpace β] {f : β → α} {b : β} :
ContinuousAt f b ↔ Tendsto (fun x => (f x, f b)) (𝓝 b) (𝓤 α) := by
rw [ContinuousAt, tendsto_nhds_left] | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | continuousAt_iff'_left | null |
continuousAt_iff_prod [TopologicalSpace β] {f : β → α} {b : β} :
ContinuousAt f b ↔ Tendsto (fun x : β × β => (f x.1, f x.2)) (𝓝 (b, b)) (𝓤 α) :=
⟨fun H => le_trans (H.prodMap' H) (nhds_le_uniformity _), fun H =>
continuousAt_iff'_left.2 <| H.comp <| tendsto_id.prodMk_nhds tendsto_const_nhds⟩ | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | continuousAt_iff_prod | null |
continuousWithinAt_iff'_right [TopologicalSpace β] {f : β → α} {b : β} {s : Set β} :
ContinuousWithinAt f s b ↔ Tendsto (fun x => (f b, f x)) (𝓝[s] b) (𝓤 α) := by
rw [ContinuousWithinAt, tendsto_nhds_right] | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | continuousWithinAt_iff'_right | null |
continuousWithinAt_iff'_left [TopologicalSpace β] {f : β → α} {b : β} {s : Set β} :
ContinuousWithinAt f s b ↔ Tendsto (fun x => (f x, f b)) (𝓝[s] b) (𝓤 α) := by
rw [ContinuousWithinAt, tendsto_nhds_left] | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | continuousWithinAt_iff'_left | null |
continuousOn_iff'_right [TopologicalSpace β] {f : β → α} {s : Set β} :
ContinuousOn f s ↔ ∀ b ∈ s, Tendsto (fun x => (f b, f x)) (𝓝[s] b) (𝓤 α) := by
simp [ContinuousOn, continuousWithinAt_iff'_right] | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | continuousOn_iff'_right | null |
continuousOn_iff'_left [TopologicalSpace β] {f : β → α} {s : Set β} :
ContinuousOn f s ↔ ∀ b ∈ s, Tendsto (fun x => (f x, f b)) (𝓝[s] b) (𝓤 α) := by
simp [ContinuousOn, continuousWithinAt_iff'_left] | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | continuousOn_iff'_left | null |
continuous_iff'_right [TopologicalSpace β] {f : β → α} :
Continuous f ↔ ∀ b, Tendsto (fun x => (f b, f x)) (𝓝 b) (𝓤 α) :=
continuous_iff_continuousAt.trans <| forall_congr' fun _ => tendsto_nhds_right | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | continuous_iff'_right | null |
continuous_iff'_left [TopologicalSpace β] {f : β → α} :
Continuous f ↔ ∀ b, Tendsto (fun x => (f x, f b)) (𝓝 b) (𝓤 α) :=
continuous_iff_continuousAt.trans <| forall_congr' fun _ => tendsto_nhds_left | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | continuous_iff'_left | null |
exists_is_open_mem_uniformity_of_forall_mem_eq
[TopologicalSpace β] {r : Set (α × α)} {s : Set β}
{f g : β → α} (hf : ∀ x ∈ s, ContinuousAt f x) (hg : ∀ x ∈ s, ContinuousAt g x)
(hfg : s.EqOn f g) (hr : r ∈ 𝓤 α) :
∃ t, IsOpen t ∧ s ⊆ t ∧ ∀ x ∈ t, (f x, g x) ∈ r := by
have A : ∀ x ∈ s, ∃ t, IsOpen t ∧ x ∈ t ∧ ∀ z ∈ t, (f z, g z) ∈ r := by
intro x hx
obtain ⟨t, ht, htsymm, htr⟩ := comp_symm_mem_uniformity_sets hr
have A : {z | (f x, f z) ∈ t} ∈ 𝓝 x := (hf x hx).preimage_mem_nhds (mem_nhds_left (f x) ht)
have B : {z | (g x, g z) ∈ t} ∈ 𝓝 x := (hg x hx).preimage_mem_nhds (mem_nhds_left (g x) ht)
rcases _root_.mem_nhds_iff.1 (inter_mem A B) with ⟨u, hu, u_open, xu⟩
refine ⟨u, u_open, xu, fun y hy ↦ ?_⟩
have I1 : (f y, f x) ∈ t := (htsymm.mk_mem_comm).2 (hu hy).1
have I2 : (g x, g y) ∈ t := (hu hy).2
rw [hfg hx] at I1
exact htr (prodMk_mem_compRel I1 I2)
choose! t t_open xt ht using A
refine ⟨⋃ x ∈ s, t x, isOpen_biUnion t_open, fun x hx ↦ mem_biUnion hx (xt x hx), ?_⟩
rintro x hx
simp only [mem_iUnion, exists_prop] at hx
rcases hx with ⟨y, ys, hy⟩
exact ht y ys x hy | lemma | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | exists_is_open_mem_uniformity_of_forall_mem_eq | Consider two functions `f` and `g` which coincide on a set `s` and are continuous there.
Then there is an open neighborhood of `s` on which `f` and `g` are uniformly close. |
Filter.Tendsto.congr_uniformity {α β} [UniformSpace β] {f g : α → β} {l : Filter α} {b : β}
(hf : Tendsto f l (𝓝 b)) (hg : Tendsto (fun x => (f x, g x)) l (𝓤 β)) : Tendsto g l (𝓝 b) :=
Uniform.tendsto_nhds_right.2 <| (Uniform.tendsto_nhds_right.1 hf).uniformity_trans hg | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | Filter.Tendsto.congr_uniformity | null |
Uniform.tendsto_congr {α β} [UniformSpace β] {f g : α → β} {l : Filter α} {b : β}
(hfg : Tendsto (fun x => (f x, g x)) l (𝓤 β)) : Tendsto f l (𝓝 b) ↔ Tendsto g l (𝓝 b) :=
⟨fun h => h.congr_uniformity hfg, fun h => h.congr_uniformity hfg.uniformity_symm⟩ | theorem | Topology | [
"Mathlib.Data.Rel",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.UniformSpace.Defs",
"Mathlib.Topology.ContinuousOn"
] | Mathlib/Topology/UniformSpace/Basic.lean | Uniform.tendsto_congr | null |
Cauchy (f : Filter α) :=
NeBot f ∧ f ×ˢ f ≤ 𝓤 α | def | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | Cauchy | A filter `f` is Cauchy if for every entourage `r`, there exists an
`s ∈ f` such that `s × s ⊆ r`. This is a generalization of Cauchy
sequences, because if `a : ℕ → α` then the filter of sets containing
cofinitely many of the `a n` is Cauchy iff `a` is a Cauchy sequence. |
IsComplete (s : Set α) :=
∀ f, Cauchy f → f ≤ 𝓟 s → ∃ x ∈ s, f ≤ 𝓝 x | def | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | IsComplete | A set `s` is called *complete*, if any Cauchy filter `f` such that `s ∈ f`
has a limit in `s` (formally, it satisfies `f ≤ 𝓝 x` for some `x ∈ s`). |
Filter.HasBasis.cauchy_iff {ι} {p : ι → Prop} {s : ι → Set (α × α)} (h : (𝓤 α).HasBasis p s)
{f : Filter α} :
Cauchy f ↔ NeBot f ∧ ∀ i, p i → ∃ t ∈ f, ∀ x ∈ t, ∀ y ∈ t, (x, y) ∈ s i :=
and_congr Iff.rfl <|
(f.basis_sets.prod_self.le_basis_iff h).trans <| by
simp only [subset_def, Prod.forall, mem_prod_eq, and_imp, id, forall_mem_comm] | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | Filter.HasBasis.cauchy_iff | null |
cauchy_iff' {f : Filter α} :
Cauchy f ↔ NeBot f ∧ ∀ s ∈ 𝓤 α, ∃ t ∈ f, ∀ x ∈ t, ∀ y ∈ t, (x, y) ∈ s :=
(𝓤 α).basis_sets.cauchy_iff | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchy_iff' | null |
cauchy_iff {f : Filter α} : Cauchy f ↔ NeBot f ∧ ∀ s ∈ 𝓤 α, ∃ t ∈ f, t ×ˢ t ⊆ s :=
cauchy_iff'.trans <| by
simp only [subset_def, Prod.forall, mem_prod_eq, and_imp, forall_mem_comm] | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchy_iff | null |
cauchy_iff_le {l : Filter α} [hl : l.NeBot] :
Cauchy l ↔ l ×ˢ l ≤ 𝓤 α := by
simp only [Cauchy, hl, true_and] | lemma | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchy_iff_le | null |
Cauchy.ultrafilter_of {l : Filter α} (h : Cauchy l) :
Cauchy (@Ultrafilter.of _ l h.1 : Filter α) := by
haveI := h.1
have := Ultrafilter.of_le l
exact ⟨Ultrafilter.neBot _, (Filter.prod_mono this this).trans h.2⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | Cauchy.ultrafilter_of | null |
cauchy_map_iff {l : Filter β} {f : β → α} :
Cauchy (l.map f) ↔ NeBot l ∧ Tendsto (fun p : β × β => (f p.1, f p.2)) (l ×ˢ l) (𝓤 α) := by
rw [Cauchy, map_neBot_iff, prod_map_map_eq, Tendsto] | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchy_map_iff | null |
cauchy_map_iff' {l : Filter β} [hl : NeBot l] {f : β → α} :
Cauchy (l.map f) ↔ Tendsto (fun p : β × β => (f p.1, f p.2)) (l ×ˢ l) (𝓤 α) :=
cauchy_map_iff.trans <| and_iff_right hl | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchy_map_iff' | null |
Cauchy.mono {f g : Filter α} [hg : NeBot g] (h_c : Cauchy f) (h_le : g ≤ f) : Cauchy g :=
⟨hg, le_trans (Filter.prod_mono h_le h_le) h_c.right⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | Cauchy.mono | null |
Cauchy.mono' {f g : Filter α} (h_c : Cauchy f) (_ : NeBot g) (h_le : g ≤ f) : Cauchy g :=
h_c.mono h_le | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | Cauchy.mono' | null |
cauchy_nhds {a : α} : Cauchy (𝓝 a) :=
⟨nhds_neBot, nhds_prod_eq.symm.trans_le (nhds_le_uniformity a)⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchy_nhds | null |
cauchy_pure {a : α} : Cauchy (pure a) :=
cauchy_nhds.mono (pure_le_nhds a) | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchy_pure | null |
Filter.Tendsto.cauchy_map {l : Filter β} [NeBot l] {f : β → α} {a : α}
(h : Tendsto f l (𝓝 a)) : Cauchy (map f l) :=
cauchy_nhds.mono h | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | Filter.Tendsto.cauchy_map | null |
Cauchy.mono_uniformSpace {u v : UniformSpace β} {F : Filter β} (huv : u ≤ v)
(hF : Cauchy (uniformSpace := u) F) : Cauchy (uniformSpace := v) F :=
⟨hF.1, hF.2.trans huv⟩ | lemma | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | Cauchy.mono_uniformSpace | null |
cauchy_inf_uniformSpace {u v : UniformSpace β} {F : Filter β} :
Cauchy (uniformSpace := u ⊓ v) F ↔
Cauchy (uniformSpace := u) F ∧ Cauchy (uniformSpace := v) F := by
unfold Cauchy
rw [inf_uniformity (u := u), le_inf_iff, and_and_left] | lemma | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchy_inf_uniformSpace | null |
cauchy_iInf_uniformSpace {ι : Sort*} [Nonempty ι] {u : ι → UniformSpace β}
{l : Filter β} :
Cauchy (uniformSpace := ⨅ i, u i) l ↔ ∀ i, Cauchy (uniformSpace := u i) l := by
unfold Cauchy
rw [iInf_uniformity, le_iInf_iff, forall_and, forall_const] | lemma | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchy_iInf_uniformSpace | null |
cauchy_iInf_uniformSpace' {ι : Sort*} {u : ι → UniformSpace β}
{l : Filter β} [l.NeBot] :
Cauchy (uniformSpace := ⨅ i, u i) l ↔ ∀ i, Cauchy (uniformSpace := u i) l := by
simp_rw [cauchy_iff_le (uniformSpace := _), iInf_uniformity, le_iInf_iff] | lemma | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchy_iInf_uniformSpace' | null |
cauchy_comap_uniformSpace {u : UniformSpace β} {α} {f : α → β} {l : Filter α} :
Cauchy (uniformSpace := comap f u) l ↔ Cauchy (map f l) := by
simp only [Cauchy, map_neBot_iff, prod_map_map_eq, map_le_iff_le_comap]
rfl | lemma | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchy_comap_uniformSpace | null |
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