statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 10
values | symbolic_name stringlengths 1 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 0 10.2k | source_url stringclasses 1
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dense_inducing_coe : dense_inducing (coe : α → completion α) | { dense := dense_range_coe,
..(uniform_inducing_coe α).inducing } | lemma | uniform_space.completion.dense_inducing_coe | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"dense",
"dense_inducing",
"inducing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_completion.complete_equiv_self [complete_space α] [separated_space α]:
completion α ≃ᵤ α | abstract_completion.compare_equiv completion.cpkg abstract_completion.of_complete | def | uniform_space.completion.uniform_completion.complete_equiv_self | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"abstract_completion.compare_equiv",
"abstract_completion.of_complete",
"complete_space",
"separated_space"
] | The uniform bijection between a complete space and its uniform completion. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separable_space_completion [separable_space α] : separable_space (completion α) | completion.dense_inducing_coe.separable_space | instance | uniform_space.completion.separable_space_completion | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_embedding_coe [separated_space α]: dense_embedding (coe : α → completion α) | { inj := separated_pure_cauchy_injective,
..dense_inducing_coe } | lemma | uniform_space.completion.dense_embedding_coe | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"dense_embedding",
"separated_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_range_coe₂ :
dense_range (λx:α × β, ((x.1 : completion α), (x.2 : completion β))) | dense_range_coe.prod_map dense_range_coe | lemma | uniform_space.completion.dense_range_coe₂ | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"dense_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_range_coe₃ :
dense_range (λx:α × (β × γ),
((x.1 : completion α), ((x.2.1 : completion β), (x.2.2 : completion γ)))) | dense_range_coe.prod_map dense_range_coe₂ | lemma | uniform_space.completion.dense_range_coe₃ | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"dense_range"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on {p : completion α → Prop}
(a : completion α) (hp : is_closed {a | p a}) (ih : ∀a:α, p a) : p a | is_closed_property dense_range_coe hp ih a | lemma | uniform_space.completion.induction_on | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"ih",
"is_closed",
"is_closed_property"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on₂ {p : completion α → completion β → Prop}
(a : completion α) (b : completion β)
(hp : is_closed {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, from
is_closed_property dense_range_coe₂ hp $ assume ⟨a, b⟩, ih a b,
this (a, b) | lemma | uniform_space.completion.induction_on₂ | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"ih",
"is_closed",
"is_closed_property"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
induction_on₃ {p : completion α → completion β → completion γ → Prop}
(a : completion α) (b : completion β) (c : completion γ)
(hp : is_closed {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, from
is_closed_property dense_range_coe₃ hp $ assume ⟨a, b, c⟩, ih a b c,
this (a, b, c) | lemma | uniform_space.completion.induction_on₃ | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"ih",
"is_closed",
"is_closed_property"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {Y : Type*} [topological_space Y] [t2_space 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 | lemma | uniform_space.completion.ext | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"continuous",
"t2_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext' {Y : Type*} [topological_space Y] [t2_space 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 | lemma | uniform_space.completion.ext' | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"continuous",
"t2_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension (f : α → β) : completion α → β | cpkg.extend f | def | uniform_space.completion.extension | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [] | "Extension" to the completion. It is defined for any map `f` but
returns an arbitrary constant value if `f` is not uniformly continuous | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_extension : uniform_continuous (completion.extension f) | cpkg.uniform_continuous_extend | lemma | uniform_space.completion.uniform_continuous_extension | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_extension : continuous (completion.extension f) | cpkg.continuous_extend | lemma | uniform_space.completion.continuous_extension | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension_coe [separated_space β] (hf : uniform_continuous f) (a : α) :
(completion.extension f) a = f a | cpkg.extend_coe hf a | lemma | uniform_space.completion.extension_coe | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"separated_space",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension_unique (hf : uniform_continuous f) {g : completion α → β}
(hg : uniform_continuous g) (h : ∀ a : α, f a = g (a : completion α)) :
completion.extension f = g | cpkg.extend_unique hf hg h | lemma | uniform_space.completion.extension_unique | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension_comp_coe {f : completion α → β} (hf : uniform_continuous f) :
completion.extension (f ∘ coe) = f | cpkg.extend_comp_coe hf | lemma | uniform_space.completion.extension_comp_coe | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : α → β) : completion α → completion β | cpkg.map cpkg f | def | uniform_space.completion.map | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [] | Completion functor acting on morphisms | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_map : uniform_continuous (completion.map f) | cpkg.uniform_continuous_map cpkg f | lemma | uniform_space.completion.uniform_continuous_map | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_map : continuous (completion.map f) | cpkg.continuous_map cpkg f | lemma | uniform_space.completion.continuous_map | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"continuous",
"continuous_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_coe (hf : uniform_continuous f) (a : α) : (completion.map f) a = f a | cpkg.map_coe cpkg hf a | lemma | uniform_space.completion.map_coe | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_unique {f : α → β} {g : completion α → completion β}
(hg : uniform_continuous g) (h : ∀a:α, ↑(f a) = g a) : completion.map f = g | cpkg.map_unique cpkg hg h | lemma | uniform_space.completion.map_unique | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id : completion.map (@id α) = id | cpkg.map_id | lemma | uniform_space.completion.map_id | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension_map [complete_space γ] [separated_space γ] {f : β → γ} {g : α → β}
(hf : uniform_continuous f) (hg : uniform_continuous g) :
completion.extension f ∘ completion.map g = completion.extension (f ∘ g) | completion.ext (continuous_extension.comp continuous_map) continuous_extension $
by intro a; simp only [hg, hf, hf.comp hg, (∘), map_coe, extension_coe] | lemma | uniform_space.completion.extension_map | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"complete_space",
"continuous_map",
"separated_space",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp {g : β → γ} {f : α → β} (hg : uniform_continuous g) (hf : uniform_continuous f) :
completion.map g ∘ completion.map f = completion.map (g ∘ f) | extension_map ((uniform_continuous_coe _).comp hg) hf | lemma | uniform_space.completion.map_comp | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"map_comp",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
completion_separation_quotient_equiv (α : Type u) [uniform_space α] :
completion (separation_quotient α) ≃ completion α | begin
refine ⟨completion.extension (separation_quotient.lift (coe : α → completion α)),
completion.map quotient.mk, _, _⟩,
{ assume a,
refine induction_on a (is_closed_eq (continuous_map.comp continuous_extension) continuous_id) _,
rintros ⟨a⟩,
show completion.map quotient.mk
(completion.exten... | def | uniform_space.completion.completion_separation_quotient_equiv | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"continuous_id",
"continuous_map",
"continuous_map.comp",
"is_closed_eq",
"separation_quotient",
"separation_quotient.lift",
"separation_quotient.lift_mk",
"uniform_space"
] | The isomorphism between the completion of a uniform space and the completion of its separation
quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_completion_separation_quotient_equiv :
uniform_continuous ⇑(completion_separation_quotient_equiv α) | uniform_continuous_extension | lemma | uniform_space.completion.uniform_continuous_completion_separation_quotient_equiv | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_completion_separation_quotient_equiv_symm :
uniform_continuous ⇑(completion_separation_quotient_equiv α).symm | uniform_continuous_map | lemma | uniform_space.completion.uniform_continuous_completion_separation_quotient_equiv_symm | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
extension₂ (f : α → β → γ) : completion α → completion β → γ | cpkg.extend₂ cpkg f | def | uniform_space.completion.extension₂ | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [] | Extend a two variable map to the Hausdorff completions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
extension₂_coe_coe (hf : uniform_continuous₂ f) (a : α) (b : β) :
completion.extension₂ f a b = f a b | cpkg.extension₂_coe_coe cpkg hf a b | lemma | uniform_space.completion.extension₂_coe_coe | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"uniform_continuous₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_extension₂ : uniform_continuous₂ (completion.extension₂ f) | cpkg.uniform_continuous_extension₂ cpkg f | lemma | uniform_space.completion.uniform_continuous_extension₂ | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"uniform_continuous₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂ (f : α → β → γ) : completion α → completion β → completion γ | cpkg.map₂ cpkg cpkg f | def | uniform_space.completion.map₂ | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [] | Lift a two variable map to the Hausdorff completions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_map₂ (f : α → β → γ) : uniform_continuous₂ (completion.map₂ f) | cpkg.uniform_continuous_map₂ cpkg cpkg f | lemma | uniform_space.completion.uniform_continuous_map₂ | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"uniform_continuous₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_map₂ {δ} [topological_space δ] {f : α → β → γ}
{a : δ → completion α} {b : δ → completion β} (ha : continuous a) (hb : continuous b) :
continuous (λd:δ, completion.map₂ f (a d) (b d)) | cpkg.continuous_map₂ cpkg cpkg ha hb | lemma | uniform_space.completion.continuous_map₂ | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"continuous",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map₂_coe_coe (a : α) (b : β) (f : α → β → γ) (hf : uniform_continuous₂ f) :
completion.map₂ f (a : completion α) (b : completion β) = f a b | cpkg.map₂_coe_coe cpkg cpkg a b f hf | lemma | uniform_space.completion.map₂_coe_coe | topology.uniform_space | src/topology/uniform_space/completion.lean | [
"topology.uniform_space.abstract_completion"
] | [
"uniform_continuous₂"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equicontinuous_at (F : ι → X → α) (x₀ : X) : Prop | ∀ U ∈ 𝓤 α, ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ U | def | equicontinuous_at | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [] | A family `F : ι → X → α` of functions from a topological space to a uniform space is
*equicontinuous at `x₀ : X`* if, for all entourage `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/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.equicontinuous_at (H : set $ X → α) (x₀ : X) : Prop | equicontinuous_at (coe : H → X → α) x₀ | abbreviation | set.equicontinuous_at | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"equicontinuous_at"
] | We say that a set `H : set (X → α)` of functions is equicontinuous at a point if the family
`coe : ↥H → (X → α)` is equicontinuous at that point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equicontinuous (F : ι → X → α) : Prop | ∀ x₀, equicontinuous_at F x₀ | def | equicontinuous | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"equicontinuous_at"
] | 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/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.equicontinuous (H : set $ X → α) : Prop | equicontinuous (coe : H → X → α) | abbreviation | set.equicontinuous | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"equicontinuous"
] | We say that a set `H : set (X → α)` of functions is equicontinuous if the family
`coe : ↥H → (X → α)` is equicontinuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_equicontinuous (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 | uniform_equicontinuous | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [] | A family `F : ι → β → α` of functions between uniform spaces is *uniformly equicontinuous* if,
for all entourage `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 x₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.uniform_equicontinuous (H : set $ β → α) : Prop | uniform_equicontinuous (coe : H → β → α) | abbreviation | set.uniform_equicontinuous | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"uniform_equicontinuous"
] | We say that a set `H : set (X → α)` of functions is uniformly equicontinuous if the family
`coe : ↥H → (X → α)` is uniformly equicontinuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equicontinuous_at_iff_pair {F : ι → X → α} {x₀ : X} : equicontinuous_at F x₀ ↔
∀ U ∈ 𝓤 α, ∃ V ∈ 𝓝 x₀, ∀ (x y ∈ V) i, (F i x, F i y) ∈ U | begin
split; intros H U hU,
{ rcases comp_symm_mem_uniformity_sets hU with ⟨V, hV, hVsymm, hVU⟩,
refine ⟨_, H V hV, λ x hx y hy i, hVU (prod_mk_mem_comp_rel _ (hy i))⟩,
exact hVsymm.mk_mem_comm.mp (hx i) },
{ rcases H U hU with ⟨V, hV, hVU⟩,
filter_upwards [hV] using λ x hx i, (hVU x₀ (mem_of_mem_nhds... | lemma | equicontinuous_at_iff_pair | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"comp_symm_mem_uniformity_sets",
"equicontinuous_at",
"mem_of_mem_nhds",
"prod_mk_mem_comp_rel"
] | Reformulation of equicontinuity at `x₀` comparing two variables near `x₀` instead of comparing
only one with `x₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_equicontinuous.equicontinuous {F : ι → β → α} (h : uniform_equicontinuous F) :
equicontinuous F | λ x₀ U hU, mem_of_superset (ball_mem_nhds x₀ (h U hU)) (λ x hx i, hx i) | lemma | uniform_equicontinuous.equicontinuous | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"equicontinuous",
"uniform_equicontinuous"
] | Uniform equicontinuity implies equicontinuity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equicontinuous_at.continuous_at {F : ι → X → α} {x₀ : X} (h : equicontinuous_at F x₀)
(i : ι) : continuous_at (F i) x₀ | begin
intros U hU,
rw uniform_space.mem_nhds_iff at hU,
rcases hU with ⟨V, hV₁, hV₂⟩,
exact mem_map.mpr (mem_of_superset (h V hV₁) (λ x hx, hV₂ (hx i)))
end | lemma | equicontinuous_at.continuous_at | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"continuous_at",
"equicontinuous_at",
"uniform_space.mem_nhds_iff"
] | Each function of a family equicontinuous at `x₀` is continuous at `x₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.equicontinuous_at.continuous_at_of_mem {H : set $ X → α} {x₀ : X}
(h : H.equicontinuous_at x₀) {f : X → α} (hf : f ∈ H) : continuous_at f x₀ | h.continuous_at ⟨f, hf⟩ | lemma | set.equicontinuous_at.continuous_at_of_mem | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"continuous_at"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equicontinuous.continuous {F : ι → X → α} (h : equicontinuous F) (i : ι) :
continuous (F i) | continuous_iff_continuous_at.mpr (λ x, (h x).continuous_at i) | lemma | equicontinuous.continuous | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"continuous",
"continuous_at",
"equicontinuous"
] | Each function of an equicontinuous family is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.equicontinuous.continuous_of_mem {H : set $ X → α} (h : H.equicontinuous)
{f : X → α} (hf : f ∈ H) : continuous f | h.continuous ⟨f, hf⟩ | lemma | set.equicontinuous.continuous_of_mem | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_equicontinuous.uniform_continuous {F : ι → β → α} (h : uniform_equicontinuous F)
(i : ι) : uniform_continuous (F i) | λ U hU, mem_map.mpr (mem_of_superset (h U hU) $ λ xy hxy, (hxy i)) | lemma | uniform_equicontinuous.uniform_continuous | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"uniform_continuous",
"uniform_equicontinuous"
] | Each function of a uniformly equicontinuous family is uniformly continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.uniform_equicontinuous.uniform_continuous_of_mem {H : set $ β → α}
(h : H.uniform_equicontinuous) {f : β → α} (hf : f ∈ H) : uniform_continuous f | h.uniform_continuous ⟨f, hf⟩ | lemma | set.uniform_equicontinuous.uniform_continuous_of_mem | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equicontinuous_at.comp {F : ι → X → α} {x₀ : X} (h : equicontinuous_at F x₀) (u : κ → ι) :
equicontinuous_at (F ∘ u) x₀ | λ U hU, (h U hU).mono (λ x H k, H (u k)) | lemma | equicontinuous_at.comp | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"equicontinuous_at"
] | Taking sub-families preserves equicontinuity at a point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.equicontinuous_at.mono {H H' : set $ X → α} {x₀ : X}
(h : H.equicontinuous_at x₀) (hH : H' ⊆ H) : H'.equicontinuous_at x₀ | h.comp (inclusion hH) | lemma | set.equicontinuous_at.mono | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equicontinuous.comp {F : ι → X → α} (h : equicontinuous F) (u : κ → ι) :
equicontinuous (F ∘ u) | λ x, (h x).comp u | lemma | equicontinuous.comp | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"equicontinuous"
] | Taking sub-families preserves equicontinuity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.equicontinuous.mono {H H' : set $ X → α}
(h : H.equicontinuous) (hH : H' ⊆ H) : H'.equicontinuous | h.comp (inclusion hH) | lemma | set.equicontinuous.mono | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_equicontinuous.comp {F : ι → β → α} (h : uniform_equicontinuous F) (u : κ → ι) :
uniform_equicontinuous (F ∘ u) | λ U hU, (h U hU).mono (λ x H k, H (u k)) | lemma | uniform_equicontinuous.comp | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"uniform_equicontinuous"
] | Taking sub-families preserves uniform equicontinuity. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set.uniform_equicontinuous.mono {H H' : set $ β → α}
(h : H.uniform_equicontinuous) (hH : H' ⊆ H) : H'.uniform_equicontinuous | h.comp (inclusion hH) | lemma | set.uniform_equicontinuous.mono | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equicontinuous_at_iff_range {F : ι → X → α} {x₀ : X} :
equicontinuous_at F x₀ ↔ equicontinuous_at (coe : range F → X → α) x₀ | ⟨λ h, by rw ← comp_range_splitting F; exact h.comp _, λ h, h.comp (range_factorization F)⟩ | lemma | equicontinuous_at_iff_range | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"equicontinuous_at"
] | A family `𝓕 : ι → X → α` is equicontinuous at `x₀` iff `range 𝓕` is equicontinuous at `x₀`,
i.e the family `coe : range F → X → α` is equicontinuous at `x₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equicontinuous_iff_range {F : ι → X → α} :
equicontinuous F ↔ equicontinuous (coe : range F → X → α) | forall_congr (λ x₀, equicontinuous_at_iff_range) | lemma | equicontinuous_iff_range | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"equicontinuous",
"equicontinuous_at_iff_range"
] | A family `𝓕 : ι → X → α` is equicontinuous iff `range 𝓕` is equicontinuous,
i.e the family `coe : range F → X → α` is equicontinuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_equicontinuous_at_iff_range {F : ι → β → α} :
uniform_equicontinuous F ↔ uniform_equicontinuous (coe : range F → β → α) | ⟨λ h, by rw ← comp_range_splitting F; exact h.comp _, λ h, h.comp (range_factorization F)⟩ | lemma | uniform_equicontinuous_at_iff_range | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"uniform_equicontinuous"
] | A family `𝓕 : ι → β → α` is uniformly equicontinuous iff `range 𝓕` is uniformly equicontinuous,
i.e the family `coe : range F → β → α` is uniformly equicontinuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equicontinuous_at_iff_continuous_at {F : ι → X → α} {x₀ : X} :
equicontinuous_at F x₀ ↔ continuous_at (of_fun ∘ function.swap F : X → ι →ᵤ α) x₀ | by rw [continuous_at, (uniform_fun.has_basis_nhds ι α _).tendsto_right_iff]; refl | lemma | equicontinuous_at_iff_continuous_at | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"continuous_at",
"equicontinuous_at",
"uniform_fun.has_basis_nhds"
] | A family `𝓕 : ι → X → α` is equicontinuous at `x₀` iff the function `swap 𝓕 : X → ι → α` is
continuous at `x₀` *when `ι → α` is equipped with the topology of uniform convergence*. This is
very useful for developping the equicontinuity API, but it should not be used directly for other
purposes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equicontinuous_iff_continuous {F : ι → X → α} :
equicontinuous F ↔ continuous (of_fun ∘ function.swap F : X → ι →ᵤ α) | by simp_rw [equicontinuous, continuous_iff_continuous_at, equicontinuous_at_iff_continuous_at] | lemma | equicontinuous_iff_continuous | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"continuous",
"continuous_iff_continuous_at",
"equicontinuous",
"equicontinuous_at_iff_continuous_at"
] | A family `𝓕 : ι → X → α` is equicontinuous iff the function `swap 𝓕 : X → ι → α` is
continuous *when `ι → α` is equipped with the topology of uniform convergence*. This is
very useful for developping the equicontinuity API, but it should not be used directly for other
purposes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_equicontinuous_iff_uniform_continuous {F : ι → β → α} :
uniform_equicontinuous F ↔ uniform_continuous (of_fun ∘ function.swap F : β → ι →ᵤ α) | by rw [uniform_continuous, (uniform_fun.has_basis_uniformity ι α).tendsto_right_iff]; refl | lemma | uniform_equicontinuous_iff_uniform_continuous | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"uniform_continuous",
"uniform_equicontinuous",
"uniform_fun.has_basis_uniformity"
] | A family `𝓕 : ι → β → α` is uniformly equicontinuous iff the function `swap 𝓕 : β → ι → α` is
uniformly continuous *when `ι → α` is equipped with the uniform structure of uniform convergence*.
This is very useful for developping the equicontinuity API, but it should not be used directly
for other purposes. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.has_basis.equicontinuous_at_iff_left {κ : Type*} {p : κ → Prop} {s : κ → set X}
{F : ι → X → α} {x₀ : X} (hX : (𝓝 x₀).has_basis p s) : equicontinuous_at F x₀ ↔
∀ U ∈ 𝓤 α, ∃ k (_ : p k), ∀ x ∈ s k, ∀ i, (F i x₀, F i x) ∈ U | begin
rw [equicontinuous_at_iff_continuous_at, continuous_at,
hX.tendsto_iff (uniform_fun.has_basis_nhds ι α _)],
refl
end | lemma | filter.has_basis.equicontinuous_at_iff_left | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"continuous_at",
"equicontinuous_at",
"equicontinuous_at_iff_continuous_at",
"uniform_fun.has_basis_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.equicontinuous_at_iff_right {κ : Type*} {p : κ → Prop} {s : κ → set (α × α)}
{F : ι → X → α} {x₀ : X} (hα : (𝓤 α).has_basis p s) : equicontinuous_at F x₀ ↔
∀ k, p k → ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ s k | begin
rw [equicontinuous_at_iff_continuous_at, continuous_at,
(uniform_fun.has_basis_nhds_of_basis ι α _ hα).tendsto_right_iff],
refl
end | lemma | filter.has_basis.equicontinuous_at_iff_right | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"continuous_at",
"equicontinuous_at",
"equicontinuous_at_iff_continuous_at",
"uniform_fun.has_basis_nhds_of_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.equicontinuous_at_iff {κ₁ κ₂ : Type*} {p₁ : κ₁ → Prop} {s₁ : κ₁ → set X}
{p₂ : κ₂ → Prop} {s₂ : κ₂ → set (α × α)} {F : ι → X → α} {x₀ : X}
(hX : (𝓝 x₀).has_basis p₁ s₁) (hα : (𝓤 α).has_basis p₂ s₂) : equicontinuous_at F x₀ ↔
∀ k₂, p₂ k₂ → ∃ k₁ (_ : p₁ k₁), ∀ x ∈ s₁ k₁, ∀ i, (F i x₀, F i x) ∈ s₂... | begin
rw [equicontinuous_at_iff_continuous_at, continuous_at,
hX.tendsto_iff (uniform_fun.has_basis_nhds_of_basis ι α _ hα)],
refl
end | lemma | filter.has_basis.equicontinuous_at_iff | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"continuous_at",
"equicontinuous_at",
"equicontinuous_at_iff_continuous_at",
"uniform_fun.has_basis_nhds_of_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.uniform_equicontinuous_iff_left {κ : Type*} {p : κ → Prop}
{s : κ → set (β × β)} {F : ι → β → α} (hβ : (𝓤 β).has_basis p s) : uniform_equicontinuous F ↔
∀ U ∈ 𝓤 α, ∃ k (_ : p k), ∀ x y, (x, y) ∈ s k → ∀ i, (F i x, F i y) ∈ U | begin
rw [uniform_equicontinuous_iff_uniform_continuous, uniform_continuous,
hβ.tendsto_iff (uniform_fun.has_basis_uniformity ι α)],
simp_rw [prod.forall],
refl
end | lemma | filter.has_basis.uniform_equicontinuous_iff_left | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"uniform_continuous",
"uniform_equicontinuous",
"uniform_equicontinuous_iff_uniform_continuous",
"uniform_fun.has_basis_uniformity"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.uniform_equicontinuous_iff_right {κ : Type*} {p : κ → Prop}
{s : κ → set (α × α)} {F : ι → β → α} (hα : (𝓤 α).has_basis p s) : uniform_equicontinuous F ↔
∀ k, p k → ∀ᶠ (xy : β × β) in 𝓤 β, ∀ i, (F i xy.1, F i xy.2) ∈ s k | begin
rw [uniform_equicontinuous_iff_uniform_continuous, uniform_continuous,
(uniform_fun.has_basis_uniformity_of_basis ι α hα).tendsto_right_iff],
refl
end | lemma | filter.has_basis.uniform_equicontinuous_iff_right | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"uniform_continuous",
"uniform_equicontinuous",
"uniform_equicontinuous_iff_uniform_continuous",
"uniform_fun.has_basis_uniformity_of_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.uniform_equicontinuous_iff {κ₁ κ₂ : Type*} {p₁ : κ₁ → Prop}
{s₁ : κ₁ → set (β × β)} {p₂ : κ₂ → Prop} {s₂ : κ₂ → set (α × α)} {F : ι → β → α}
(hβ : (𝓤 β).has_basis p₁ s₁) (hα : (𝓤 α).has_basis p₂ s₂) : uniform_equicontinuous F ↔
∀ k₂, p₂ k₂ → ∃ k₁ (_ : p₁ k₁), ∀ x y, (x, y) ∈ s₁ k₁ → ∀ i, (F i x... | begin
rw [uniform_equicontinuous_iff_uniform_continuous, uniform_continuous,
hβ.tendsto_iff (uniform_fun.has_basis_uniformity_of_basis ι α hα)],
simp_rw [prod.forall],
refl
end | lemma | filter.has_basis.uniform_equicontinuous_iff | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"uniform_continuous",
"uniform_equicontinuous",
"uniform_equicontinuous_iff_uniform_continuous",
"uniform_fun.has_basis_uniformity_of_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_inducing.equicontinuous_at_iff {F : ι → X → α} {x₀ : X} {u : α → β}
(hu : uniform_inducing u) :
equicontinuous_at F x₀ ↔ equicontinuous_at (((∘) u) ∘ F) x₀ | begin
have := (uniform_fun.postcomp_uniform_inducing hu).inducing,
rw [equicontinuous_at_iff_continuous_at, equicontinuous_at_iff_continuous_at,
this.continuous_at_iff],
refl
end | lemma | uniform_inducing.equicontinuous_at_iff | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"equicontinuous_at",
"equicontinuous_at_iff_continuous_at",
"inducing",
"uniform_fun.postcomp_uniform_inducing",
"uniform_inducing"
] | Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous at a point
`x₀ : X` iff the family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is
equicontinuous at `x₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing.equicontinuous_iff {F : ι → X → α} {u : α → β}
(hu : uniform_inducing u) :
equicontinuous F ↔ equicontinuous (((∘) u) ∘ F) | begin
congrm (∀ x, (_ : Prop)),
rw hu.equicontinuous_at_iff
end | lemma | uniform_inducing.equicontinuous_iff | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"equicontinuous",
"uniform_inducing"
] | Given `u : α → β` a uniform inducing map, a family `𝓕 : ι → X → α` is equicontinuous iff the
family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is equicontinuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing.uniform_equicontinuous_iff {F : ι → β → α} {u : α → γ}
(hu : uniform_inducing u) :
uniform_equicontinuous F ↔ uniform_equicontinuous (((∘) u) ∘ F) | begin
have := uniform_fun.postcomp_uniform_inducing hu,
rw [uniform_equicontinuous_iff_uniform_continuous, uniform_equicontinuous_iff_uniform_continuous,
this.uniform_continuous_iff],
refl
end | lemma | uniform_inducing.uniform_equicontinuous_iff | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"uniform_equicontinuous",
"uniform_equicontinuous_iff_uniform_continuous",
"uniform_fun.postcomp_uniform_inducing",
"uniform_inducing"
] | Given `u : α → γ` a uniform inducing map, a family `𝓕 : ι → β → α` is uniformly equicontinuous
iff the family `𝓕'`, obtained by precomposing each function of `𝓕` by `u`, is uniformly
equicontinuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equicontinuous_at.closure' {A : set Y} {u : Y → X → α} {x₀ : X}
(hA : equicontinuous_at (u ∘ coe : A → X → α) x₀) (hu : continuous u) :
equicontinuous_at (u ∘ coe : closure A → X → α) x₀ | begin
intros U hU,
rcases mem_uniformity_is_closed hU with ⟨V, hV, hVclosed, hVU⟩,
filter_upwards [hA V hV] with x hx,
rw set_coe.forall at *,
change A ⊆ (λ f, (u f x₀, u f x)) ⁻¹' V at hx,
refine (closure_minimal hx $ hVclosed.preimage $ _).trans (preimage_mono hVU),
exact continuous.prod_mk ((continuous... | lemma | equicontinuous_at.closure' | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"closure",
"closure_minimal",
"continuous",
"continuous.prod_mk",
"continuous_apply",
"equicontinuous_at",
"mem_uniformity_is_closed",
"set_coe.forall"
] | A version of `equicontinuous_at.closure` applicable to subsets of types which embed continuously
into `X → α` with the product topology. It turns out we don't need any other condition on the
embedding than continuity, but in practice this will mostly be applied to `fun_like` types where
the coercion is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equicontinuous_at.closure {A : set $ X → α} {x₀ : X} (hA : A.equicontinuous_at x₀) :
(closure A).equicontinuous_at x₀ | @equicontinuous_at.closure' _ _ _ _ _ _ _ id _ hA continuous_id | lemma | equicontinuous_at.closure | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"closure",
"continuous_id",
"equicontinuous_at",
"equicontinuous_at.closure'"
] | If a set of functions is equicontinuous at some `x₀`, its closure for the product topology is
also equicontinuous at `x₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.continuous_at_of_equicontinuous_at {l : filter ι} [l.ne_bot] {F : ι → X → α}
{f : X → α} {x₀ : X} (h₁ : tendsto F l (𝓝 f)) (h₂ : equicontinuous_at F x₀) :
continuous_at f x₀ | (equicontinuous_at_iff_range.mp h₂).closure.continuous_at
⟨f, mem_closure_of_tendsto h₁ $ eventually_of_forall mem_range_self⟩ | lemma | filter.tendsto.continuous_at_of_equicontinuous_at | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"continuous_at",
"equicontinuous_at",
"filter",
"mem_closure_of_tendsto"
] | If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise* along some nontrivial filter, and if the
family `𝓕` is equicontinuous at some `x₀ : X`, then the limit is continuous at `x₀`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equicontinuous.closure' {A : set Y} {u : Y → X → α}
(hA : equicontinuous (u ∘ coe : A → X → α)) (hu : continuous u) :
equicontinuous (u ∘ coe : closure A → X → α) | λ x, (hA x).closure' hu | lemma | equicontinuous.closure' | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"closure",
"continuous",
"equicontinuous"
] | A version of `equicontinuous.closure` applicable to subsets of types which embed continuously
into `X → α` with the product topology. It turns out we don't need any other condition on the
embedding than continuity, but in practice this will mostly be applied to `fun_like` types where
the coercion is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equicontinuous.closure {A : set $ X → α} (hA : A.equicontinuous) :
(closure A).equicontinuous | λ x, (hA x).closure | lemma | equicontinuous.closure | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"closure",
"equicontinuous"
] | If a set of functions is equicontinuous, its closure for the product topology is also
equicontinuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.continuous_of_equicontinuous_at {l : filter ι} [l.ne_bot] {F : ι → X → α}
{f : X → α} (h₁ : tendsto F l (𝓝 f)) (h₂ : equicontinuous F) :
continuous f | continuous_iff_continuous_at.mpr (λ x, h₁.continuous_at_of_equicontinuous_at (h₂ x)) | lemma | filter.tendsto.continuous_of_equicontinuous_at | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"continuous",
"equicontinuous",
"filter"
] | If `𝓕 : ι → X → α` tends to `f : X → α` *pointwise* along some nontrivial filter, and if the
family `𝓕` is equicontinuous, then the limit is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_equicontinuous.closure' {A : set Y} {u : Y → β → α}
(hA : uniform_equicontinuous (u ∘ coe : A → β → α)) (hu : continuous u) :
uniform_equicontinuous (u ∘ coe : closure A → β → α) | begin
intros U hU,
rcases mem_uniformity_is_closed hU with ⟨V, hV, hVclosed, hVU⟩,
filter_upwards [hA V hV],
rintros ⟨x, y⟩ hxy,
rw set_coe.forall at *,
change A ⊆ (λ f, (u f x, u f y)) ⁻¹' V at hxy,
refine (closure_minimal hxy $ hVclosed.preimage $ _).trans (preimage_mono hVU),
exact continuous.prod_mk... | lemma | uniform_equicontinuous.closure' | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"closure",
"closure_minimal",
"continuous",
"continuous.prod_mk",
"continuous_apply",
"mem_uniformity_is_closed",
"set_coe.forall",
"uniform_equicontinuous"
] | A version of `uniform_equicontinuous.closure` applicable to subsets of types which embed
continuously into `β → α` with the product topology. It turns out we don't need any other condition
on the embedding than continuity, but in practice this will mostly be applied to `fun_like` types
where the coercion is injective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_equicontinuous.closure {A : set $ β → α} (hA : A.uniform_equicontinuous) :
(closure A).uniform_equicontinuous | @uniform_equicontinuous.closure' _ _ _ _ _ _ _ id hA continuous_id | lemma | uniform_equicontinuous.closure | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"closure",
"continuous_id",
"uniform_equicontinuous",
"uniform_equicontinuous.closure'"
] | If a set of functions is uniformly equicontinuous, its closure for the product topology is also
uniformly equicontinuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.tendsto.uniform_continuous_of_uniform_equicontinuous {l : filter ι} [l.ne_bot]
{F : ι → β → α} {f : β → α} (h₁ : tendsto F l (𝓝 f)) (h₂ : uniform_equicontinuous F) :
uniform_continuous f | (uniform_equicontinuous_at_iff_range.mp h₂).closure.uniform_continuous
⟨f, mem_closure_of_tendsto h₁ $ eventually_of_forall mem_range_self⟩ | lemma | filter.tendsto.uniform_continuous_of_uniform_equicontinuous | topology.uniform_space | src/topology/uniform_space/equicontinuity.lean | [
"topology.uniform_space.uniform_convergence_topology"
] | [
"filter",
"mem_closure_of_tendsto",
"uniform_continuous",
"uniform_equicontinuous"
] | If `𝓕 : ι → β → α` tends to `f : β → α` *pointwise* along some nontrivial filter, and if the
family `𝓕` is uniformly equicontinuous, then the limit is uniformly continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_equiv (α : Type*) (β : Type*) [uniform_space α] [uniform_space β]
extends α ≃ β | (uniform_continuous_to_fun : uniform_continuous to_fun)
(uniform_continuous_inv_fun : uniform_continuous inv_fun) | structure | uniform_equiv | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"inv_fun",
"uniform_continuous",
"uniform_space"
] | Uniform isomorphism between `α` and `β` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_equiv_mk_coe (a : equiv α β) (b c) :
((uniform_equiv.mk a b c) : α → β) = a | rfl | lemma | uniform_equiv.uniform_equiv_mk_coe | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm (h : α ≃ᵤ β) : β ≃ᵤ α | { uniform_continuous_to_fun := h.uniform_continuous_inv_fun,
uniform_continuous_inv_fun := h.uniform_continuous_to_fun,
to_equiv := h.to_equiv.symm } | def | uniform_equiv.symm | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | Inverse of a uniform isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simps.apply (h : α ≃ᵤ β) : α → β | h | def | uniform_equiv.simps.apply | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | See Note [custom simps projection]. We need to specify this projection explicitly in this case,
because it is a composition of multiple projections. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simps.symm_apply (h : α ≃ᵤ β) : β → α | h.symm
initialize_simps_projections uniform_equiv
(to_equiv_to_fun → apply, to_equiv_inv_fun → symm_apply, -to_equiv) | def | uniform_equiv.simps.symm_apply | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"uniform_equiv"
] | See Note [custom simps projection] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_equiv (h : α ≃ᵤ β) : ⇑h.to_equiv = h | rfl | lemma | uniform_equiv.coe_to_equiv | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_symm_to_equiv (h : α ≃ᵤ β) : ⇑h.to_equiv.symm = h.symm | rfl | lemma | uniform_equiv.coe_symm_to_equiv | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_equiv_injective : function.injective (to_equiv : α ≃ᵤ β → α ≃ β) | | ⟨e, h₁, h₂⟩ ⟨e', h₁', h₂'⟩ rfl := rfl | lemma | uniform_equiv.to_equiv_injective | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {h h' : α ≃ᵤ β} (H : ∀ x, h x = h' x) : h = h' | to_equiv_injective $ equiv.ext H | lemma | uniform_equiv.ext | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"equiv.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl (α : Type*) [uniform_space α] : α ≃ᵤ α | { uniform_continuous_to_fun := uniform_continuous_id,
uniform_continuous_inv_fun := uniform_continuous_id,
to_equiv := equiv.refl α } | def | uniform_equiv.refl | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"equiv.refl",
"uniform_continuous_id",
"uniform_space"
] | Identity map as a uniform isomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans (h₁ : α ≃ᵤ β) (h₂ : β ≃ᵤ γ) : α ≃ᵤ γ | { uniform_continuous_to_fun := h₂.uniform_continuous_to_fun.comp h₁.uniform_continuous_to_fun,
uniform_continuous_inv_fun := h₁.uniform_continuous_inv_fun.comp h₂.uniform_continuous_inv_fun,
to_equiv := equiv.trans h₁.to_equiv h₂.to_equiv } | def | uniform_equiv.trans | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"equiv.trans"
] | Composition of two uniform isomorphisms. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
trans_apply (h₁ : α ≃ᵤ β) (h₂ : β ≃ᵤ γ) (a : α) : h₁.trans h₂ a = h₂ (h₁ a) | rfl | lemma | uniform_equiv.trans_apply | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_equiv_mk_coe_symm (a : equiv α β) (b c) :
((uniform_equiv.mk a b c).symm : β → α) = a.symm | rfl | lemma | uniform_equiv.uniform_equiv_mk_coe_symm | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"equiv"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
refl_symm : (uniform_equiv.refl α).symm = uniform_equiv.refl α | rfl | lemma | uniform_equiv.refl_symm | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"uniform_equiv.refl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous (h : α ≃ᵤ β) : uniform_continuous h | h.uniform_continuous_to_fun | lemma | uniform_equiv.uniform_continuous | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous (h : α ≃ᵤ β) : continuous h | h.uniform_continuous.continuous | lemma | uniform_equiv.continuous | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_symm (h : α ≃ᵤ β) : uniform_continuous (h.symm) | h.uniform_continuous_inv_fun | lemma | uniform_equiv.uniform_continuous_symm | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_symm (h : α ≃ᵤ β) : continuous (h.symm) | h.uniform_continuous_symm.continuous | lemma | uniform_equiv.continuous_symm | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_homeomorph (e : α ≃ᵤ β) : α ≃ₜ β | { continuous_to_fun := e.continuous,
continuous_inv_fun := e.continuous_symm,
.. e.to_equiv } | def | uniform_equiv.to_homeomorph | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | A uniform isomorphism as a homeomorphism. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
apply_symm_apply (h : α ≃ᵤ β) (x : β) : h (h.symm x) = x | h.to_equiv.apply_symm_apply x | lemma | uniform_equiv.apply_symm_apply | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
symm_apply_apply (h : α ≃ᵤ β) (x : α) : h.symm (h x) = x | h.to_equiv.symm_apply_apply x | lemma | uniform_equiv.symm_apply_apply | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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