fact stringlengths 6 14.3k | statement stringlengths 1 2.88k | proof stringlengths 0 13.9k | type stringclasses 12
values | symbolic_name stringlengths 0 131 | library stringclasses 417
values | filename stringlengths 17 80 | imports listlengths 0 16 | deps listlengths 0 64 | docstring stringlengths 8 10.2k ⌀ | line_start int64 6 4.24k | line_end int64 7 4.25k | has_proof bool 2
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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 | 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`. | 360 | 362 | true | 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_... | 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`. | 366 | 370 | true | 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 | 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. | 376 | 379 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equicontinuous.closure {A : set $ X → α} (hA : A.equicontinuous) :
(closure A).equicontinuous :=
λ x, (hA x).closure | 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. | 383 | 385 | true | 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)) | 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. | 389 | 392 | true | 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 ⟨... | 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. | 398 | 410 | true | 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 | 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. | 414 | 416 | true | 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... | 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. | 420 | 424 | true | 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) | 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` | 34 | 38 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: has_coe_to_fun (α ≃ᵤ β) (λ _, α → β) := ⟨λe, e.to_equiv⟩ | : has_coe_to_fun (α ≃ᵤ β) (λ _, α → β) | ⟨λe, e.to_equiv⟩ | instance | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | null | 45 | 45 | true | 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 | 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"
] | null | 47 | 49 | true | 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 } | 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. | 52 | 55 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
simps.apply (h : α ≃ᵤ β) : α → β := h | 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. | 59 | 59 | true | 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) | 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] | 61 | 64 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_to_equiv (h : α ≃ᵤ β) : ⇑h.to_equiv = h := rfl | 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"
] | [] | null | 66 | 66 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_symm_to_equiv (h : α ≃ᵤ β) : ⇑h.to_equiv.symm = h.symm := rfl | 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"
] | [] | null | 67 | 67 | true | 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 | 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"
] | [] | null | 69 | 70 | true | 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 | 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"
] | null | 72 | 73 | true | 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 } | 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. | 76 | 80 | true | 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 } | 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. | 83 | 86 | true | 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 | 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"
] | [] | null | 88 | 88 | true | 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 | 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"
] | null | 90 | 92 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
refl_symm : (uniform_equiv.refl α).symm = uniform_equiv.refl α := rfl | 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"
] | null | 94 | 94 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous (h : α ≃ᵤ β) : uniform_continuous h :=
h.uniform_continuous_to_fun | 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"
] | null | 96 | 97 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous (h : α ≃ᵤ β) : continuous h :=
h.uniform_continuous.continuous | 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"
] | null | 99 | 101 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_symm (h : α ≃ᵤ β) : uniform_continuous (h.symm) :=
h.uniform_continuous_inv_fun | 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"
] | null | 103 | 104 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
continuous_symm (h : α ≃ᵤ β) : continuous (h.symm) :=
h.uniform_continuous_symm.continuous | 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"
] | null | 106 | 108 | true | 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 } | 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. | 111 | 115 | true | 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 | 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"
] | [] | null | 117 | 118 | true | 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 | 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"
] | [] | null | 120 | 121 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
bijective (h : α ≃ᵤ β) : function.bijective h := h.to_equiv.bijective | bijective (h : α ≃ᵤ β) : function.bijective h | h.to_equiv.bijective | lemma | uniform_equiv.bijective | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | null | 123 | 123 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
injective (h : α ≃ᵤ β) : function.injective h := h.to_equiv.injective | injective (h : α ≃ᵤ β) : function.injective h | h.to_equiv.injective | lemma | uniform_equiv.injective | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | null | 124 | 124 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
surjective (h : α ≃ᵤ β) : function.surjective h := h.to_equiv.surjective | surjective (h : α ≃ᵤ β) : function.surjective h | h.to_equiv.surjective | lemma | uniform_equiv.surjective | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | null | 125 | 125 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
change_inv (f : α ≃ᵤ β) (g : β → α) (hg : function.right_inverse g f) : α ≃ᵤ β :=
have g = f.symm, from funext (λ x, calc g x = f.symm (f (g x)) : (f.left_inv (g x)).symm
... = f.symm x : by rw hg x),
{ to_fun := f,
inv_fun := g,
left_inv := by convert f.left_inv,
right_inv... | change_inv (f : α ≃ᵤ β) (g : β → α) (hg : function.right_inverse g f) : α ≃ᵤ β | have g = f.symm, from funext (λ x, calc g x = f.symm (f (g x)) : (f.left_inv (g x)).symm
... = f.symm x : by rw hg x),
{ to_fun := f,
inv_fun := g,
left_inv := by convert f.left_inv,
right_inv := by convert f.right_inv,
uniform_continuous_to_fun := f.uniform_continuous,
... | def | uniform_equiv.change_inv | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"inv_fun"
] | Change the uniform equiv `f` to make the inverse function definitionally equal to `g`. | 128 | 136 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
symm_comp_self (h : α ≃ᵤ β) : ⇑h.symm ∘ ⇑h = id :=
funext h.symm_apply_apply | symm_comp_self (h : α ≃ᵤ β) : ⇑h.symm ∘ ⇑h = id | funext h.symm_apply_apply | lemma | uniform_equiv.symm_comp_self | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | null | 138 | 139 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
self_comp_symm (h : α ≃ᵤ β) : ⇑h ∘ ⇑h.symm = id :=
funext h.apply_symm_apply | self_comp_symm (h : α ≃ᵤ β) : ⇑h ∘ ⇑h.symm = id | funext h.apply_symm_apply | lemma | uniform_equiv.self_comp_symm | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | null | 141 | 142 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
range_coe (h : α ≃ᵤ β) : range h = univ :=
h.surjective.range_eq | range_coe (h : α ≃ᵤ β) : range h = univ | h.surjective.range_eq | lemma | uniform_equiv.range_coe | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | null | 144 | 145 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_symm (h : α ≃ᵤ β) : image h.symm = preimage h :=
funext h.symm.to_equiv.image_eq_preimage | image_symm (h : α ≃ᵤ β) : image h.symm = preimage h | funext h.symm.to_equiv.image_eq_preimage | lemma | uniform_equiv.image_symm | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | null | 147 | 148 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preimage_symm (h : α ≃ᵤ β) : preimage h.symm = image h :=
(funext h.to_equiv.image_eq_preimage).symm | preimage_symm (h : α ≃ᵤ β) : preimage h.symm = image h | (funext h.to_equiv.image_eq_preimage).symm | lemma | uniform_equiv.preimage_symm | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | null | 150 | 151 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image_preimage (h : α ≃ᵤ β) (s : set β) : h '' (h ⁻¹' s) = s :=
h.to_equiv.image_preimage s | image_preimage (h : α ≃ᵤ β) (s : set β) : h '' (h ⁻¹' s) = s | h.to_equiv.image_preimage s | lemma | uniform_equiv.image_preimage | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | null | 153 | 154 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preimage_image (h : α ≃ᵤ β) (s : set α) : h ⁻¹' (h '' s) = s :=
h.to_equiv.preimage_image s | preimage_image (h : α ≃ᵤ β) (s : set α) : h ⁻¹' (h '' s) = s | h.to_equiv.preimage_image s | lemma | uniform_equiv.preimage_image | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | null | 156 | 157 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_inducing (h : α ≃ᵤ β) : uniform_inducing h :=
uniform_inducing_of_compose h.uniform_continuous h.symm.uniform_continuous $
by simp only [symm_comp_self, uniform_inducing_id] | uniform_inducing (h : α ≃ᵤ β) : uniform_inducing h | uniform_inducing_of_compose h.uniform_continuous h.symm.uniform_continuous $
by simp only [symm_comp_self, uniform_inducing_id] | lemma | uniform_equiv.uniform_inducing | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"uniform_inducing",
"uniform_inducing_id",
"uniform_inducing_of_compose"
] | null | 159 | 161 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_eq (h : α ≃ᵤ β) : uniform_space.comap h ‹_› = ‹_› :=
by ext : 1; exact h.uniform_inducing.comap_uniformity | comap_eq (h : α ≃ᵤ β) : uniform_space.comap h ‹_› = ‹_› | by ext : 1; exact h.uniform_inducing.comap_uniformity | lemma | uniform_equiv.comap_eq | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"uniform_space.comap"
] | null | 163 | 164 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_embedding (h : α ≃ᵤ β) : uniform_embedding h :=
⟨h.uniform_inducing, h.injective⟩ | uniform_embedding (h : α ≃ᵤ β) : uniform_embedding h | ⟨h.uniform_inducing, h.injective⟩ | lemma | uniform_equiv.uniform_embedding | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"uniform_embedding"
] | null | 166 | 167 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_uniform_embedding (f : α → β) (hf : uniform_embedding f) :
α ≃ᵤ (set.range f) :=
{ uniform_continuous_to_fun := hf.to_uniform_inducing.uniform_continuous.subtype_mk _,
uniform_continuous_inv_fun :=
by simp [hf.to_uniform_inducing.uniform_continuous_iff, uniform_continuous_subtype_coe],
to_equiv := equiv.of... | of_uniform_embedding (f : α → β) (hf : uniform_embedding f) :
α ≃ᵤ (set.range f) | { uniform_continuous_to_fun := hf.to_uniform_inducing.uniform_continuous.subtype_mk _,
uniform_continuous_inv_fun :=
by simp [hf.to_uniform_inducing.uniform_continuous_iff, uniform_continuous_subtype_coe],
to_equiv := equiv.of_injective f hf.inj } | def | uniform_equiv.of_uniform_embedding | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"equiv.of_injective",
"set.range",
"uniform_continuous_subtype_coe",
"uniform_embedding"
] | Uniform equiv given a uniform embedding. | 170 | 175 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
set_congr {s t : set α} (h : s = t) : s ≃ᵤ t :=
{ uniform_continuous_to_fun := uniform_continuous_subtype_val.subtype_mk _,
uniform_continuous_inv_fun := uniform_continuous_subtype_val.subtype_mk _,
to_equiv := equiv.set_congr h } | set_congr {s t : set α} (h : s = t) : s ≃ᵤ t | { uniform_continuous_to_fun := uniform_continuous_subtype_val.subtype_mk _,
uniform_continuous_inv_fun := uniform_continuous_subtype_val.subtype_mk _,
to_equiv := equiv.set_congr h } | def | uniform_equiv.set_congr | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"equiv.set_congr"
] | If two sets are equal, then they are uniformly equivalent. | 178 | 181 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_congr (h₁ : α ≃ᵤ β) (h₂ : γ ≃ᵤ δ) : α × γ ≃ᵤ β × δ :=
{ uniform_continuous_to_fun := (h₁.uniform_continuous.comp uniform_continuous_fst).prod_mk
(h₂.uniform_continuous.comp uniform_continuous_snd),
uniform_continuous_inv_fun := (h₁.symm.uniform_continuous.comp uniform_continuous_fst).prod_mk
(h₂.symm.un... | prod_congr (h₁ : α ≃ᵤ β) (h₂ : γ ≃ᵤ δ) : α × γ ≃ᵤ β × δ | { uniform_continuous_to_fun := (h₁.uniform_continuous.comp uniform_continuous_fst).prod_mk
(h₂.uniform_continuous.comp uniform_continuous_snd),
uniform_continuous_inv_fun := (h₁.symm.uniform_continuous.comp uniform_continuous_fst).prod_mk
(h₂.symm.uniform_continuous.comp uniform_continuous_snd),
to_equiv :... | def | uniform_equiv.prod_congr | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"uniform_continuous_fst",
"uniform_continuous_snd"
] | Product of two uniform isomorphisms. | 184 | 189 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_congr_symm (h₁ : α ≃ᵤ β) (h₂ : γ ≃ᵤ δ) :
(h₁.prod_congr h₂).symm = h₁.symm.prod_congr h₂.symm := rfl | prod_congr_symm (h₁ : α ≃ᵤ β) (h₂ : γ ≃ᵤ δ) :
(h₁.prod_congr h₂).symm = h₁.symm.prod_congr h₂.symm | rfl | lemma | uniform_equiv.prod_congr_symm | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | null | 191 | 192 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod_congr (h₁ : α ≃ᵤ β) (h₂ : γ ≃ᵤ δ) :
⇑(h₁.prod_congr h₂) = prod.map h₁ h₂ := rfl | coe_prod_congr (h₁ : α ≃ᵤ β) (h₂ : γ ≃ᵤ δ) :
⇑(h₁.prod_congr h₂) = prod.map h₁ h₂ | rfl | lemma | uniform_equiv.coe_prod_congr | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | null | 194 | 195 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_comm : α × β ≃ᵤ β × α :=
{ uniform_continuous_to_fun := uniform_continuous_snd.prod_mk uniform_continuous_fst,
uniform_continuous_inv_fun := uniform_continuous_snd.prod_mk uniform_continuous_fst,
to_equiv := equiv.prod_comm α β } | prod_comm : α × β ≃ᵤ β × α | { uniform_continuous_to_fun := uniform_continuous_snd.prod_mk uniform_continuous_fst,
uniform_continuous_inv_fun := uniform_continuous_snd.prod_mk uniform_continuous_fst,
to_equiv := equiv.prod_comm α β } | def | uniform_equiv.prod_comm | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"equiv.prod_comm",
"uniform_continuous_fst"
] | `α × β` is uniformly isomorphic to `β × α`. | 201 | 204 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_comm_symm : (prod_comm α β).symm = prod_comm β α := rfl | prod_comm_symm : (prod_comm α β).symm = prod_comm β α | rfl | lemma | uniform_equiv.prod_comm_symm | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | null | 206 | 206 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod_comm : ⇑(prod_comm α β) = prod.swap := rfl | coe_prod_comm : ⇑(prod_comm α β) = prod.swap | rfl | lemma | uniform_equiv.coe_prod_comm | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"prod.swap"
] | null | 207 | 207 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_assoc : (α × β) × γ ≃ᵤ α × (β × γ) :=
{ uniform_continuous_to_fun := (uniform_continuous_fst.comp uniform_continuous_fst).prod_mk
((uniform_continuous_snd.comp uniform_continuous_fst).prod_mk uniform_continuous_snd),
uniform_continuous_inv_fun := (uniform_continuous_fst.prod_mk
(uniform_continuous_fst.c... | prod_assoc : (α × β) × γ ≃ᵤ α × (β × γ) | { uniform_continuous_to_fun := (uniform_continuous_fst.comp uniform_continuous_fst).prod_mk
((uniform_continuous_snd.comp uniform_continuous_fst).prod_mk uniform_continuous_snd),
uniform_continuous_inv_fun := (uniform_continuous_fst.prod_mk
(uniform_continuous_fst.comp uniform_continuous_snd)).prod_mk
(u... | def | uniform_equiv.prod_assoc | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"equiv.prod_assoc",
"uniform_continuous_fst",
"uniform_continuous_snd"
] | `(α × β) × γ` is uniformly isomorphic to `α × (β × γ)`. | 210 | 216 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod_punit : α × punit ≃ᵤ α :=
{ to_equiv := equiv.prod_punit α,
uniform_continuous_to_fun := uniform_continuous_fst,
uniform_continuous_inv_fun := uniform_continuous_id.prod_mk uniform_continuous_const } | prod_punit : α × punit ≃ᵤ α | { to_equiv := equiv.prod_punit α,
uniform_continuous_to_fun := uniform_continuous_fst,
uniform_continuous_inv_fun := uniform_continuous_id.prod_mk uniform_continuous_const } | def | uniform_equiv.prod_punit | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"equiv.prod_punit",
"uniform_continuous_const",
"uniform_continuous_fst"
] | `α × {*}` is uniformly isomorphic to `α`. | 219 | 223 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
punit_prod : punit × α ≃ᵤ α :=
(prod_comm _ _).trans (prod_punit _) | punit_prod : punit × α ≃ᵤ α | (prod_comm _ _).trans (prod_punit _) | def | uniform_equiv.punit_prod | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | `{*} × α` is uniformly isomorphic to `α`. | 226 | 227 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_punit_prod : ⇑(punit_prod α) = prod.snd := rfl | coe_punit_prod : ⇑(punit_prod α) = prod.snd | rfl | lemma | uniform_equiv.coe_punit_prod | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [] | null | 229 | 229 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ulift : ulift.{v u} α ≃ᵤ α :=
{ uniform_continuous_to_fun := uniform_continuous_comap,
uniform_continuous_inv_fun := begin
have hf : uniform_inducing (@equiv.ulift.{v u} α).to_fun, from ⟨rfl⟩,
simp_rw [hf.uniform_continuous_iff],
exact uniform_continuous_id,
end,
.. equiv.ulift } | ulift : ulift.{v u} α ≃ᵤ α | { uniform_continuous_to_fun := uniform_continuous_comap,
uniform_continuous_inv_fun := begin
have hf : uniform_inducing (@equiv.ulift.{v u} α).to_fun, from ⟨rfl⟩,
simp_rw [hf.uniform_continuous_iff],
exact uniform_continuous_id,
end,
.. equiv.ulift } | def | uniform_equiv.ulift | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"equiv.ulift",
"uniform_continuous_comap",
"uniform_continuous_id",
"uniform_inducing"
] | Uniform equivalence between `ulift α` and `α`. | 232 | 239 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fun_unique (ι α : Type*) [unique ι] [uniform_space α] : (ι → α) ≃ᵤ α :=
{ to_equiv := equiv.fun_unique ι α,
uniform_continuous_to_fun := Pi.uniform_continuous_proj _ _,
uniform_continuous_inv_fun := uniform_continuous_pi.mpr (λ _, uniform_continuous_id) } | fun_unique (ι α : Type*) [unique ι] [uniform_space α] : (ι → α) ≃ᵤ α | { to_equiv := equiv.fun_unique ι α,
uniform_continuous_to_fun := Pi.uniform_continuous_proj _ _,
uniform_continuous_inv_fun := uniform_continuous_pi.mpr (λ _, uniform_continuous_id) } | def | uniform_equiv.fun_unique | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"Pi.uniform_continuous_proj",
"equiv.fun_unique",
"uniform_continuous_id",
"uniform_space",
"unique"
] | If `ι` has a unique element, then `ι → α` is homeomorphic to `α`. | 244 | 248 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pi_fin_two (α : fin 2 → Type u) [Π i, uniform_space (α i)] : (Π i, α i) ≃ᵤ α 0 × α 1 :=
{ to_equiv := pi_fin_two_equiv α,
uniform_continuous_to_fun :=
(Pi.uniform_continuous_proj _ 0).prod_mk (Pi.uniform_continuous_proj _ 1),
uniform_continuous_inv_fun := uniform_continuous_pi.mpr $
fin.forall_fin_two.2 ⟨un... | pi_fin_two (α : fin 2 → Type u) [Π i, uniform_space (α i)] : (Π i, α i) ≃ᵤ α 0 × α 1 | { to_equiv := pi_fin_two_equiv α,
uniform_continuous_to_fun :=
(Pi.uniform_continuous_proj _ 0).prod_mk (Pi.uniform_continuous_proj _ 1),
uniform_continuous_inv_fun := uniform_continuous_pi.mpr $
fin.forall_fin_two.2 ⟨uniform_continuous_fst, uniform_continuous_snd⟩ } | def | uniform_equiv.pi_fin_two | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"Pi.uniform_continuous_proj",
"pi_fin_two_equiv",
"uniform_space"
] | Uniform isomorphism between dependent functions `Π i : fin 2, α i` and `α 0 × α 1`. | 251 | 257 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
fin_two_arrow : (fin 2 → α) ≃ᵤ α × α :=
{ to_equiv := fin_two_arrow_equiv α, .. pi_fin_two (λ _, α) } | fin_two_arrow : (fin 2 → α) ≃ᵤ α × α | { to_equiv := fin_two_arrow_equiv α, .. pi_fin_two (λ _, α) } | def | uniform_equiv.fin_two_arrow | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"fin_two_arrow_equiv"
] | Uniform isomorphism between `α² = fin 2 → α` and `α × α`. | 260 | 261 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
image (e : α ≃ᵤ β) (s : set α) : s ≃ᵤ e '' s :=
{ uniform_continuous_to_fun :=
(e.uniform_continuous.comp uniform_continuous_subtype_val).subtype_mk _,
uniform_continuous_inv_fun :=
(e.symm.uniform_continuous.comp uniform_continuous_subtype_val).subtype_mk _,
to_equiv := e.to_equiv.image s } | image (e : α ≃ᵤ β) (s : set α) : s ≃ᵤ e '' s | { uniform_continuous_to_fun :=
(e.uniform_continuous.comp uniform_continuous_subtype_val).subtype_mk _,
uniform_continuous_inv_fun :=
(e.symm.uniform_continuous.comp uniform_continuous_subtype_val).subtype_mk _,
to_equiv := e.to_equiv.image s } | def | uniform_equiv.image | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"uniform_continuous_subtype_val"
] | A subset of a uniform space is uniformly isomorphic to its image under a uniform isomorphism. | 266 | 271 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv.to_uniform_equiv_of_uniform_inducing [uniform_space α] [uniform_space β]
(f : α ≃ β) (hf : uniform_inducing f) :
α ≃ᵤ β :=
{ uniform_continuous_to_fun := hf.uniform_continuous,
uniform_continuous_inv_fun := hf.uniform_continuous_iff.2 $ by simpa using uniform_continuous_id,
.. f } | equiv.to_uniform_equiv_of_uniform_inducing [uniform_space α] [uniform_space β]
(f : α ≃ β) (hf : uniform_inducing f) :
α ≃ᵤ β | { uniform_continuous_to_fun := hf.uniform_continuous,
uniform_continuous_inv_fun := hf.uniform_continuous_iff.2 $ by simpa using uniform_continuous_id,
.. f } | def | equiv.to_uniform_equiv_of_uniform_inducing | topology.uniform_space | src/topology/uniform_space/equiv.lean | [
"topology.homeomorph",
"topology.uniform_space.uniform_embedding",
"topology.uniform_space.pi"
] | [
"uniform_continuous_id",
"uniform_inducing",
"uniform_space"
] | A uniform inducing equiv between uniform spaces is a uniform isomorphism. | 276 | 281 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
: uniform_space (matrix m n 𝕜) :=
(by apply_instance : uniform_space (m → n → 𝕜)) | : uniform_space (matrix m n 𝕜) | (by apply_instance : uniform_space (m → n → 𝕜)) | instance | topology.uniform_space | src/topology/uniform_space/matrix.lean | [
"topology.uniform_space.pi",
"data.matrix.basic"
] | [
"matrix",
"uniform_space"
] | null | 22 | 23 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniformity :
𝓤 (matrix m n 𝕜) = ⨅ (i : m) (j : n), (𝓤 𝕜).comap (λ a, (a.1 i j, a.2 i j)) :=
begin
erw [Pi.uniformity, Pi.uniformity],
simp_rw [filter.comap_infi, filter.comap_comap],
refl,
end | uniformity :
𝓤 (matrix m n 𝕜) = ⨅ (i : m) (j : n), (𝓤 𝕜).comap (λ a, (a.1 i j, a.2 i j)) | begin
erw [Pi.uniformity, Pi.uniformity],
simp_rw [filter.comap_infi, filter.comap_comap],
refl,
end | lemma | matrix.uniformity | topology.uniform_space | src/topology/uniform_space/matrix.lean | [
"topology.uniform_space.pi",
"data.matrix.basic"
] | [
"Pi.uniformity",
"filter.comap_comap",
"filter.comap_infi",
"matrix",
"uniformity"
] | null | 25 | 31 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous {β : Type*} [uniform_space β] {f : β → matrix m n 𝕜} :
uniform_continuous f ↔ ∀ i j, uniform_continuous (λ x, f x i j) :=
by simp only [uniform_continuous, matrix.uniformity, filter.tendsto_infi, filter.tendsto_comap_iff] | uniform_continuous {β : Type*} [uniform_space β] {f : β → matrix m n 𝕜} :
uniform_continuous f ↔ ∀ i j, uniform_continuous (λ x, f x i j) | by simp only [uniform_continuous, matrix.uniformity, filter.tendsto_infi, filter.tendsto_comap_iff] | lemma | matrix.uniform_continuous | topology.uniform_space | src/topology/uniform_space/matrix.lean | [
"topology.uniform_space.pi",
"data.matrix.basic"
] | [
"filter.tendsto_comap_iff",
"filter.tendsto_infi",
"matrix",
"matrix.uniformity",
"uniform_continuous",
"uniform_space"
] | null | 33 | 35 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
[complete_space 𝕜] : complete_space (matrix m n 𝕜) :=
(by apply_instance : complete_space (m → n → 𝕜)) | [complete_space 𝕜] : complete_space (matrix m n 𝕜) | (by apply_instance : complete_space (m → n → 𝕜)) | instance | topology.uniform_space | src/topology/uniform_space/matrix.lean | [
"topology.uniform_space.pi",
"data.matrix.basic"
] | [
"complete_space",
"matrix"
] | null | 37 | 38 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
[separated_space 𝕜] : separated_space (matrix m n 𝕜) :=
(by apply_instance : separated_space (m → n → 𝕜)) | [separated_space 𝕜] : separated_space (matrix m n 𝕜) | (by apply_instance : separated_space (m → n → 𝕜)) | instance | topology.uniform_space | src/topology/uniform_space/matrix.lean | [
"topology.uniform_space.pi",
"data.matrix.basic"
] | [
"matrix",
"separated_space"
] | null | 40 | 41 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Pi.uniform_space : uniform_space (Πi, α i) :=
uniform_space.of_core_eq
(⨅i, uniform_space.comap (λ a : Πi, α i, a i) (U i)).to_core
Pi.topological_space $ eq.symm to_topological_space_infi | Pi.uniform_space : uniform_space (Πi, α i) | uniform_space.of_core_eq
(⨅i, uniform_space.comap (λ a : Πi, α i, a i) (U i)).to_core
Pi.topological_space $ eq.symm to_topological_space_infi | instance | Pi.uniform_space | topology.uniform_space | src/topology/uniform_space/pi.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation"
] | [
"Pi.topological_space",
"to_topological_space_infi",
"uniform_space",
"uniform_space.comap",
"uniform_space.of_core_eq"
] | null | 26 | 29 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pi.uniformity :
𝓤 (Π i, α i) = ⨅ i : ι, filter.comap (λ a, (a.1 i, a.2 i)) $ 𝓤 (α i) :=
infi_uniformity | Pi.uniformity :
𝓤 (Π i, α i) = ⨅ i : ι, filter.comap (λ a, (a.1 i, a.2 i)) $ 𝓤 (α i) | infi_uniformity | lemma | Pi.uniformity | topology.uniform_space | src/topology/uniform_space/pi.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation"
] | [
"filter.comap",
"infi_uniformity"
] | null | 31 | 33 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_continuous_pi {β : Type*} [uniform_space β] {f : β → Π i, α i} :
uniform_continuous f ↔ ∀ i, uniform_continuous (λ x, f x i) :=
by simp only [uniform_continuous, Pi.uniformity, tendsto_infi, tendsto_comap_iff] | uniform_continuous_pi {β : Type*} [uniform_space β] {f : β → Π i, α i} :
uniform_continuous f ↔ ∀ i, uniform_continuous (λ x, f x i) | by simp only [uniform_continuous, Pi.uniformity, tendsto_infi, tendsto_comap_iff] | lemma | uniform_continuous_pi | topology.uniform_space | src/topology/uniform_space/pi.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation"
] | [
"Pi.uniformity",
"uniform_continuous",
"uniform_space"
] | null | 37 | 39 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pi.uniform_continuous_proj (i : ι) : uniform_continuous (λ (a : Π (i : ι), α i), a i) :=
uniform_continuous_pi.1 uniform_continuous_id i | Pi.uniform_continuous_proj (i : ι) : uniform_continuous (λ (a : Π (i : ι), α i), a i) | uniform_continuous_pi.1 uniform_continuous_id i | lemma | Pi.uniform_continuous_proj | topology.uniform_space | src/topology/uniform_space/pi.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation"
] | [
"uniform_continuous",
"uniform_continuous_id"
] | null | 43 | 44 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pi.complete [∀ i, complete_space (α i)] : complete_space (Π i, α i) :=
⟨begin
intros f hf,
haveI := hf.1,
have : ∀ i, ∃ x : α i, filter.map (λ a : Πi, α i, a i) f ≤ 𝓝 x,
{ intro i,
have key : cauchy (map (λ (a : Π (i : ι), α i), a i) f),
from hf.map (Pi.uniform_continuous_proj α i),
exact cauchy_... | Pi.complete [∀ i, complete_space (α i)] : complete_space (Π i, α i) | ⟨begin
intros f hf,
haveI := hf.1,
have : ∀ i, ∃ x : α i, filter.map (λ a : Πi, α i, a i) f ≤ 𝓝 x,
{ intro i,
have key : cauchy (map (λ (a : Π (i : ι), α i), a i) f),
from hf.map (Pi.uniform_continuous_proj α i),
exact cauchy_iff_exists_le_nhds.1 key },
choose x hx using this,
use x,
rwa [n... | instance | Pi.complete | topology.uniform_space | src/topology/uniform_space/pi.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation"
] | [
"Pi.uniform_continuous_proj",
"cauchy",
"complete_space",
"filter.map",
"nhds_pi"
] | null | 46 | 58 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Pi.separated [∀ i, separated_space (α i)] : separated_space (Π i, α i) :=
separated_def.2 $ assume x y H,
begin
ext i,
apply eq_of_separated_of_uniform_continuous (Pi.uniform_continuous_proj α i),
apply H,
end | Pi.separated [∀ i, separated_space (α i)] : separated_space (Π i, α i) | separated_def.2 $ assume x y H,
begin
ext i,
apply eq_of_separated_of_uniform_continuous (Pi.uniform_continuous_proj α i),
apply H,
end | instance | Pi.separated | topology.uniform_space | src/topology/uniform_space/pi.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation"
] | [
"Pi.uniform_continuous_proj",
"separated_space"
] | null | 60 | 66 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniform_space.to_regular_space : regular_space α :=
regular_space.of_basis
(λ a, by { rw [nhds_eq_comap_uniformity], exact uniformity_has_basis_closed.comap _ })
(λ a V hV, hV.2.preimage $ continuous_const.prod_mk continuous_id) | uniform_space.to_regular_space : regular_space α | regular_space.of_basis
(λ a, by { rw [nhds_eq_comap_uniformity], exact uniformity_has_basis_closed.comap _ })
(λ a V hV, hV.2.preimage $ continuous_const.prod_mk continuous_id) | instance | uniform_space.to_regular_space | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"continuous_id",
"nhds_eq_comap_uniformity",
"regular_space",
"regular_space.of_basis"
] | null | 85 | 89 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separation_rel (α : Type u) [u : uniform_space α] :=
⋂₀ (𝓤 α).sets | separation_rel (α : Type u) [u : uniform_space α] | ⋂₀ (𝓤 α).sets | def | separation_rel | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"uniform_space"
] | The separation relation is the intersection of all entourages.
Two points which are related by the separation relation are "indistinguishable"
according to the uniform structure. | 94 | 95 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separated_equiv : equivalence (λx y, (x, y) ∈ 𝓢 α) :=
⟨assume x, assume s, refl_mem_uniformity,
assume x y, assume h (s : set (α×α)) hs,
have preimage prod.swap s ∈ 𝓤 α,
from symm_le_uniformity hs,
h _ this,
assume x y z (hxy : (x, y) ∈ 𝓢 α) (hyz : (y, z) ∈ 𝓢 α)
s (hs : s ∈ 𝓤 α),
let ⟨t... | separated_equiv : equivalence (λx y, (x, y) ∈ 𝓢 α) | ⟨assume x, assume s, refl_mem_uniformity,
assume x y, assume h (s : set (α×α)) hs,
have preimage prod.swap s ∈ 𝓤 α,
from symm_le_uniformity hs,
h _ this,
assume x y z (hxy : (x, y) ∈ 𝓢 α) (hyz : (y, z) ∈ 𝓢 α)
s (hs : s ∈ 𝓤 α),
let ⟨t, ht, (h_ts : comp_rel t t ⊆ s)⟩ := comp_mem_uniformity... | lemma | separated_equiv | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"comp_mem_uniformity_sets",
"comp_rel",
"prod.swap",
"refl_mem_uniformity",
"symm_le_uniformity"
] | null | 99 | 109 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.has_basis.mem_separation_rel {ι : Sort*} {p : ι → Prop} {s : ι → set (α × α)}
(h : (𝓤 α).has_basis p s) {a : α × α} :
a ∈ 𝓢 α ↔ ∀ i, p i → a ∈ s i :=
h.forall_mem_mem | filter.has_basis.mem_separation_rel {ι : Sort*} {p : ι → Prop} {s : ι → set (α × α)}
(h : (𝓤 α).has_basis p s) {a : α × α} :
a ∈ 𝓢 α ↔ ∀ i, p i → a ∈ s i | h.forall_mem_mem | lemma | filter.has_basis.mem_separation_rel | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [] | null | 111 | 114 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separation_rel_iff_specializes {a b : α} : (a, b) ∈ 𝓢 α ↔ a ⤳ b :=
by simp only [(𝓤 α).basis_sets.mem_separation_rel, id, mem_set_of_eq,
(nhds_basis_uniformity (𝓤 α).basis_sets).specializes_iff] | separation_rel_iff_specializes {a b : α} : (a, b) ∈ 𝓢 α ↔ a ⤳ b | by simp only [(𝓤 α).basis_sets.mem_separation_rel, id, mem_set_of_eq,
(nhds_basis_uniformity (𝓤 α).basis_sets).specializes_iff] | theorem | separation_rel_iff_specializes | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"nhds_basis_uniformity"
] | null | 116 | 118 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separation_rel_iff_inseparable {a b : α} : (a, b) ∈ 𝓢 α ↔ inseparable a b :=
separation_rel_iff_specializes.trans specializes_iff_inseparable | separation_rel_iff_inseparable {a b : α} : (a, b) ∈ 𝓢 α ↔ inseparable a b | separation_rel_iff_specializes.trans specializes_iff_inseparable | theorem | separation_rel_iff_inseparable | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"inseparable",
"specializes_iff_inseparable"
] | null | 120 | 121 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separated_space (α : Type u) [uniform_space α] : Prop := (out : 𝓢 α = id_rel) | separated_space (α : Type u) [uniform_space α] : Prop | (out : 𝓢 α = id_rel) | class | separated_space | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"id_rel",
"uniform_space"
] | A uniform space is separated if its separation relation is trivial (each point
is related only to itself). | 125 | 125 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separated_space_iff {α : Type u} [uniform_space α] :
separated_space α ↔ 𝓢 α = id_rel :=
⟨λ h, h.1, λ h, ⟨h⟩⟩ | separated_space_iff {α : Type u} [uniform_space α] :
separated_space α ↔ 𝓢 α = id_rel | ⟨λ h, h.1, λ h, ⟨h⟩⟩ | theorem | separated_space_iff | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"id_rel",
"separated_space",
"uniform_space"
] | null | 127 | 129 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separated_def {α : Type u} [uniform_space α] :
separated_space α ↔ ∀ x y, (∀ r ∈ 𝓤 α, (x, y) ∈ r) → x = y :=
by simp [separated_space_iff, id_rel_subset.2 separated_equiv.1, subset.antisymm_iff];
simp [subset_def, separation_rel] | separated_def {α : Type u} [uniform_space α] :
separated_space α ↔ ∀ x y, (∀ r ∈ 𝓤 α, (x, y) ∈ r) → x = y | by simp [separated_space_iff, id_rel_subset.2 separated_equiv.1, subset.antisymm_iff];
simp [subset_def, separation_rel] | theorem | separated_def | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"separated_space",
"separated_space_iff",
"separation_rel",
"uniform_space"
] | null | 131 | 134 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separated_def' {α : Type u} [uniform_space α] :
separated_space α ↔ ∀ x y, x ≠ y → ∃ r ∈ 𝓤 α, (x, y) ∉ r :=
separated_def.trans $ forall₂_congr $ λ x y, by rw ← not_imp_not; simp [not_forall] | separated_def' {α : Type u} [uniform_space α] :
separated_space α ↔ ∀ x y, x ≠ y → ∃ r ∈ 𝓤 α, (x, y) ∉ r | separated_def.trans $ forall₂_congr $ λ x y, by rw ← not_imp_not; simp [not_forall] | theorem | separated_def' | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"forall₂_congr",
"not_forall",
"not_imp_not",
"separated_space",
"uniform_space"
] | null | 136 | 138 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_uniformity {α : Type*} [uniform_space α] [separated_space α] {x y : α}
(h : ∀ {V}, V ∈ 𝓤 α → (x, y) ∈ V) : x = y :=
separated_def.mp ‹separated_space α› x y (λ _, h) | eq_of_uniformity {α : Type*} [uniform_space α] [separated_space α] {x y : α}
(h : ∀ {V}, V ∈ 𝓤 α → (x, y) ∈ V) : x = y | separated_def.mp ‹separated_space α› x y (λ _, h) | lemma | eq_of_uniformity | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"separated_space",
"uniform_space"
] | null | 140 | 142 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_uniformity_basis {α : Type*} [uniform_space α] [separated_space α] {ι : Type*}
{p : ι → Prop} {s : ι → set (α × α)} (hs : (𝓤 α).has_basis p s) {x y : α}
(h : ∀ {i}, p i → (x, y) ∈ s i) : x = y :=
eq_of_uniformity (λ V V_in, let ⟨i, hi, H⟩ := hs.mem_iff.mp V_in in H (h hi)) | eq_of_uniformity_basis {α : Type*} [uniform_space α] [separated_space α] {ι : Type*}
{p : ι → Prop} {s : ι → set (α × α)} (hs : (𝓤 α).has_basis p s) {x y : α}
(h : ∀ {i}, p i → (x, y) ∈ s i) : x = y | eq_of_uniformity (λ V V_in, let ⟨i, hi, H⟩ := hs.mem_iff.mp V_in in H (h hi)) | lemma | eq_of_uniformity_basis | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"eq_of_uniformity",
"separated_space",
"uniform_space"
] | null | 144 | 147 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_forall_symmetric {α : Type*} [uniform_space α] [separated_space α] {x y : α}
(h : ∀ {V}, V ∈ 𝓤 α → symmetric_rel V → (x, y) ∈ V) : x = y :=
eq_of_uniformity_basis has_basis_symmetric (by simpa [and_imp] using λ _, h) | eq_of_forall_symmetric {α : Type*} [uniform_space α] [separated_space α] {x y : α}
(h : ∀ {V}, V ∈ 𝓤 α → symmetric_rel V → (x, y) ∈ V) : x = y | eq_of_uniformity_basis has_basis_symmetric (by simpa [and_imp] using λ _, h) | lemma | eq_of_forall_symmetric | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"and_imp",
"eq_of_uniformity_basis",
"separated_space",
"symmetric_rel",
"uniform_space"
] | null | 149 | 151 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
eq_of_cluster_pt_uniformity [separated_space α] {x y : α} (h : cluster_pt (x, y) (𝓤 α)) :
x = y :=
eq_of_uniformity_basis uniformity_has_basis_closed $ λ V ⟨hV, hVc⟩,
is_closed_iff_cluster_pt.1 hVc _ $ h.mono $ le_principal_iff.2 hV | eq_of_cluster_pt_uniformity [separated_space α] {x y : α} (h : cluster_pt (x, y) (𝓤 α)) :
x = y | eq_of_uniformity_basis uniformity_has_basis_closed $ λ V ⟨hV, hVc⟩,
is_closed_iff_cluster_pt.1 hVc _ $ h.mono $ le_principal_iff.2 hV | lemma | eq_of_cluster_pt_uniformity | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"cluster_pt",
"eq_of_uniformity_basis",
"separated_space",
"uniformity_has_basis_closed"
] | null | 153 | 156 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
id_rel_sub_separation_relation (α : Type*) [uniform_space α] : id_rel ⊆ 𝓢 α :=
begin
unfold separation_rel,
rw id_rel_subset,
intros x,
suffices : ∀ t ∈ 𝓤 α, (x, x) ∈ t, by simpa only [refl_mem_uniformity],
exact λ t, refl_mem_uniformity,
end | id_rel_sub_separation_relation (α : Type*) [uniform_space α] : id_rel ⊆ 𝓢 α | begin
unfold separation_rel,
rw id_rel_subset,
intros x,
suffices : ∀ t ∈ 𝓤 α, (x, x) ∈ t, by simpa only [refl_mem_uniformity],
exact λ t, refl_mem_uniformity,
end | lemma | id_rel_sub_separation_relation | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"id_rel",
"id_rel_subset",
"refl_mem_uniformity",
"separation_rel",
"uniform_space"
] | null | 158 | 165 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separation_rel_comap {f : α → β}
(h : ‹uniform_space α› = uniform_space.comap f ‹uniform_space β›) :
𝓢 α = (prod.map f f) ⁻¹' 𝓢 β :=
begin
unfreezingI { subst h },
dsimp [separation_rel],
simp_rw [uniformity_comap, (filter.comap_has_basis (prod.map f f) (𝓤 β)).sInter_sets,
← preimage_Inter, sInter_e... | separation_rel_comap {f : α → β}
(h : ‹uniform_space α› = uniform_space.comap f ‹uniform_space β›) :
𝓢 α = (prod.map f f) ⁻¹' 𝓢 β | begin
unfreezingI { subst h },
dsimp [separation_rel],
simp_rw [uniformity_comap, (filter.comap_has_basis (prod.map f f) (𝓤 β)).sInter_sets,
← preimage_Inter, sInter_eq_bInter],
refl,
end | lemma | separation_rel_comap | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"filter.comap_has_basis",
"separation_rel",
"uniform_space.comap",
"uniformity_comap"
] | null | 167 | 176 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
filter.has_basis.separation_rel {ι : Sort*} {p : ι → Prop} {s : ι → set (α × α)}
(h : has_basis (𝓤 α) p s) :
𝓢 α = ⋂ i (hi : p i), s i :=
by { unfold separation_rel, rw h.sInter_sets } | filter.has_basis.separation_rel {ι : Sort*} {p : ι → Prop} {s : ι → set (α × α)}
(h : has_basis (𝓤 α) p s) :
𝓢 α = ⋂ i (hi : p i), s i | by { unfold separation_rel, rw h.sInter_sets } | lemma | filter.has_basis.separation_rel | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"separation_rel"
] | null | 178 | 181 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separation_rel_eq_inter_closure : 𝓢 α = ⋂₀ (closure '' (𝓤 α).sets) :=
by simp [uniformity_has_basis_closure.separation_rel] | separation_rel_eq_inter_closure : 𝓢 α = ⋂₀ (closure '' (𝓤 α).sets) | by simp [uniformity_has_basis_closure.separation_rel] | lemma | separation_rel_eq_inter_closure | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"closure"
] | null | 183 | 184 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed_separation_rel : is_closed (𝓢 α) :=
begin
rw separation_rel_eq_inter_closure,
apply is_closed_sInter,
rintros _ ⟨t, t_in, rfl⟩,
exact is_closed_closure,
end | is_closed_separation_rel : is_closed (𝓢 α) | begin
rw separation_rel_eq_inter_closure,
apply is_closed_sInter,
rintros _ ⟨t, t_in, rfl⟩,
exact is_closed_closure,
end | lemma | is_closed_separation_rel | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"is_closed",
"is_closed_closure",
"is_closed_sInter",
"separation_rel_eq_inter_closure"
] | null | 186 | 192 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separated_iff_t2 : separated_space α ↔ t2_space α :=
begin
classical,
split ; introI h,
{ rw [t2_iff_is_closed_diagonal, ← show 𝓢 α = diagonal α, from h.1],
exact is_closed_separation_rel },
{ rw separated_def',
intros x y hxy,
rcases t2_separation hxy with ⟨u, v, uo, vo, hx, hy, h⟩,
rcases is_... | separated_iff_t2 : separated_space α ↔ t2_space α | begin
classical,
split ; introI h,
{ rw [t2_iff_is_closed_diagonal, ← show 𝓢 α = diagonal α, from h.1],
exact is_closed_separation_rel },
{ rw separated_def',
intros x y hxy,
rcases t2_separation hxy with ⟨u, v, uo, vo, hx, hy, h⟩,
rcases is_open_iff_ball_subset.1 uo x hx with ⟨r, hrU, hr⟩,
... | lemma | separated_iff_t2 | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"is_closed_separation_rel",
"separated_def'",
"separated_space",
"t2_iff_is_closed_diagonal",
"t2_separation",
"t2_space"
] | null | 194 | 205 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separated_t3 [separated_space α] : t3_space α :=
by { haveI := separated_iff_t2.mp ‹_›, exact ⟨⟩ } | separated_t3 [separated_space α] : t3_space α | by { haveI := separated_iff_t2.mp ‹_›, exact ⟨⟩ } | instance | separated_t3 | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"separated_space",
"t3_space"
] | null | 207 | 209 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
subtype.separated_space [separated_space α] (s : set α) : separated_space s :=
separated_iff_t2.mpr subtype.t2_space | subtype.separated_space [separated_space α] (s : set α) : separated_space s | separated_iff_t2.mpr subtype.t2_space | instance | subtype.separated_space | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"separated_space"
] | null | 211 | 212 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed_of_spaced_out [separated_space α] {V₀ : set (α × α)} (V₀_in : V₀ ∈ 𝓤 α)
{s : set α} (hs : s.pairwise (λ x y, (x, y) ∉ V₀)) : is_closed s :=
begin
rcases comp_symm_mem_uniformity_sets V₀_in with ⟨V₁, V₁_in, V₁_symm, h_comp⟩,
apply is_closed_of_closure_subset,
intros x hx,
rw mem_closure_iff_ball at ... | is_closed_of_spaced_out [separated_space α] {V₀ : set (α × α)} (V₀_in : V₀ ∈ 𝓤 α)
{s : set α} (hs : s.pairwise (λ x y, (x, y) ∉ V₀)) : is_closed s | begin
rcases comp_symm_mem_uniformity_sets V₀_in with ⟨V₁, V₁_in, V₁_symm, h_comp⟩,
apply is_closed_of_closure_subset,
intros x hx,
rw mem_closure_iff_ball at hx,
rcases hx V₁_in with ⟨y, hy, hy'⟩,
suffices : x = y, by rwa this,
apply eq_of_forall_symmetric,
intros V V_in V_symm,
rcases hx (inter_mem ... | lemma | is_closed_of_spaced_out | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"ball_inter_left",
"ball_inter_right",
"by_contra",
"comp_symm_mem_uniformity_sets",
"eq_of_forall_symmetric",
"is_closed",
"is_closed_of_closure_subset",
"mem_comp_of_mem_ball",
"separated_space"
] | null | 214 | 230 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed_range_of_spaced_out {ι} [separated_space α] {V₀ : set (α × α)} (V₀_in : V₀ ∈ 𝓤 α)
{f : ι → α} (hf : pairwise (λ x y, (f x, f y) ∉ V₀)) : is_closed (range f) :=
is_closed_of_spaced_out V₀_in $
by { rintro _ ⟨x, rfl⟩ _ ⟨y, rfl⟩ h, exact hf (ne_of_apply_ne f h) } | is_closed_range_of_spaced_out {ι} [separated_space α] {V₀ : set (α × α)} (V₀_in : V₀ ∈ 𝓤 α)
{f : ι → α} (hf : pairwise (λ x y, (f x, f y) ∉ V₀)) : is_closed (range f) | is_closed_of_spaced_out V₀_in $
by { rintro _ ⟨x, rfl⟩ _ ⟨y, rfl⟩ h, exact hf (ne_of_apply_ne f h) } | lemma | is_closed_range_of_spaced_out | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"is_closed",
"is_closed_of_spaced_out",
"ne_of_apply_ne",
"pairwise",
"separated_space"
] | null | 232 | 235 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separation_setoid (α : Type u) [uniform_space α] : setoid α :=
⟨λx y, (x, y) ∈ 𝓢 α, separated_equiv⟩ | separation_setoid (α : Type u) [uniform_space α] : setoid α | ⟨λx y, (x, y) ∈ 𝓢 α, separated_equiv⟩ | def | uniform_space.separation_setoid | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"uniform_space"
] | The separation relation of a uniform space seen as a setoid. | 244 | 245 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
separation_setoid.uniform_space {α : Type u} [u : uniform_space α] :
uniform_space (quotient (separation_setoid α)) :=
{ to_topological_space := u.to_topological_space.coinduced (λx, ⟦x⟧),
uniformity := map (λp:(α×α), (⟦p.1⟧, ⟦p.2⟧)) u.uniformity,
refl := le_trans (by simp [quotient.exists_rep]) (filter.map_mono ... | separation_setoid.uniform_space {α : Type u} [u : uniform_space α] :
uniform_space (quotient (separation_setoid α)) | { to_topological_space := u.to_topological_space.coinduced (λx, ⟦x⟧),
uniformity := map (λp:(α×α), (⟦p.1⟧, ⟦p.2⟧)) u.uniformity,
refl := le_trans (by simp [quotient.exists_rep]) (filter.map_mono refl_le_uniformity),
symm := tendsto_map' $
by simp [prod.swap, (∘)]; exact tendsto_map.comp tendsto_swap_uniformit... | instance | uniform_space.separation_setoid.uniform_space | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"comp_le_uniformity3",
"comp_mem_uniformity_sets",
"comp_rel",
"filter.map_mono",
"forall_quotient_iff",
"is_open_coinduced",
"is_open_uniformity",
"mem_map",
"monotone_id",
"prod.swap",
"quotient.eq",
"refl_le_uniformity",
"set.image",
"tendsto_swap_uniformity",
"uniform_space",
"unif... | null | 249 | 293 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
uniformity_quotient :
𝓤 (quotient (separation_setoid α)) = (𝓤 α).map (λp:(α×α), (⟦p.1⟧, ⟦p.2⟧)) :=
rfl | uniformity_quotient :
𝓤 (quotient (separation_setoid α)) = (𝓤 α).map (λp:(α×α), (⟦p.1⟧, ⟦p.2⟧)) | rfl | lemma | uniform_space.uniformity_quotient | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [] | null | 295 | 297 | true | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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