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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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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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"
] | [] | 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 := 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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"
] | [] | 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 | 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"
] | [] | 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 | 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"
] | [] | 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 | 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"
] | [] | 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 | 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"
] | [] | 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] | 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"
] | 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 | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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"
] | 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_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. | 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 } | 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. | 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.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. | 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 | 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"
] | [] | 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 | 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"
] | [] | 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 } | 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`. | 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.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)`. | 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 } | 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`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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`. | 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 } | 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`. | 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) } | 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`. | 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 ⟨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`. | 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) } | 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`. | 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 } | 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. | 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 } | 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. | 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 | 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"
] | 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] | 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"
] | 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 | 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"
] | 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 | lemma | Pi.uniformity | topology.uniform_space | src/topology/uniform_space/pi.lean | [
"topology.uniform_space.cauchy",
"topology.uniform_space.separation"
] | [
"filter.comap",
"infi_uniformity"
] | 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] | 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"
] | 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 | 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"
] | 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_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"
] | 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 | 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"
] | 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) | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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. | 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, 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"
] | 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 | 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"
] | [] | 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] | 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"
] | 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 | 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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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). | 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⟩⟩ | 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"
] | 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] | 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"
] | 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] | 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"
] | 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) | 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"
] | 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)) | 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"
] | 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) | 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"
] | 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 | 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"
] | 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 | 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"
] | 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_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"
] | 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 } | 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"
] | 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] | 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"
] | 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 | 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"
] | 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_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"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
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"
] | 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 | instance | subtype.separated_space | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"separated_space"
] | 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 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"
] | 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) } | 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"
] | 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⟩ | 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. | 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 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... | 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 | lemma | uniform_space.uniformity_quotient | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_quotient_mk :
uniform_continuous (quotient.mk : α → quotient (separation_setoid α)) | le_rfl | lemma | uniform_space.uniform_continuous_quotient_mk | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"le_rfl",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_quotient {f : quotient (separation_setoid α) → β}
(hf : uniform_continuous (λx, f ⟦x⟧)) : uniform_continuous f | hf | lemma | uniform_space.uniform_continuous_quotient | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_quotient_lift
{f : α → β} {h : ∀a b, (a, b) ∈ 𝓢 α → f a = f b}
(hf : uniform_continuous f) : uniform_continuous (λa, quotient.lift f h a) | uniform_continuous_quotient hf | lemma | uniform_space.uniform_continuous_quotient_lift | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_quotient_lift₂
{f : α → β → γ} {h : ∀a c b d, (a, b) ∈ 𝓢 α → (c, d) ∈ 𝓢 β → f a c = f b d}
(hf : uniform_continuous (λp:α×β, f p.1 p.2)) :
uniform_continuous (λp:_×_, quotient.lift₂ f h p.1 p.2) | begin
rw [uniform_continuous, uniformity_prod_eq_prod, uniformity_quotient, uniformity_quotient,
filter.prod_map_map_eq, filter.tendsto_map'_iff, filter.tendsto_map'_iff],
rwa [uniform_continuous, uniformity_prod_eq_prod, filter.tendsto_map'_iff] at hf
end | lemma | uniform_space.uniform_continuous_quotient_lift₂ | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"filter.prod_map_map_eq",
"filter.tendsto_map'_iff",
"uniform_continuous",
"uniformity_prod_eq_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_quotient_le_uniformity :
(𝓤 $ quotient $ separation_setoid α).comap (λ (p : α × α), (⟦p.fst⟧, ⟦p.snd⟧)) ≤ (𝓤 α) | assume t' ht',
let ⟨t, ht, tt_t'⟩ := comp_mem_uniformity_sets ht' in
let ⟨s, hs, ss_t⟩ := comp_mem_uniformity_sets ht in
⟨(λp:α×α, (⟦p.1⟧, ⟦p.2⟧)) '' s,
(𝓤 α).sets_of_superset hs $ assume x hx, ⟨x, hx, rfl⟩,
assume ⟨a₁, a₂⟩ ⟨⟨b₁, b₂⟩, hb, ab_eq⟩,
have ⟦b₁⟧ = ⟦a₁⟧ ∧ ⟦b₂⟧ = ⟦a₂⟧, from prod.mk.inj ab_eq,
have b₁ ... | lemma | uniform_space.comap_quotient_le_uniformity | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"comp_mem_uniformity_sets"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_quotient_eq_uniformity :
(𝓤 $ quotient $ separation_setoid α).comap (λ (p : α × α), (⟦p.fst⟧, ⟦p.snd⟧)) = 𝓤 α | le_antisymm comap_quotient_le_uniformity le_comap_map | lemma | uniform_space.comap_quotient_eq_uniformity | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
separated_separation : separated_space (quotient (separation_setoid α)) | ⟨set.ext $ assume ⟨a, b⟩, quotient.induction_on₂ a b $ assume a b,
⟨assume h,
have a ≈ b, from assume s hs,
have s ∈ (𝓤 $ quotient $ separation_setoid α).comap (λp:(α×α), (⟦p.1⟧, ⟦p.2⟧)),
from comap_quotient_le_uniformity hs,
let ⟨t, ht, hts⟩ := this in
hts begin dsimp [preimage], exact... | instance | uniform_space.separated_separation | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"refl_mem_uniformity",
"separated_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
separated_of_uniform_continuous {f : α → β} {x y : α}
(H : uniform_continuous f) (h : x ≈ y) : f x ≈ f y | assume _ h', h _ (H h') | lemma | uniform_space.separated_of_uniform_continuous | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_of_separated_of_uniform_continuous [separated_space β] {f : α → β} {x y : α}
(H : uniform_continuous f) (h : x ≈ y) : f x = f y | separated_def.1 (by apply_instance) _ _ $ separated_of_uniform_continuous H h | lemma | uniform_space.eq_of_separated_of_uniform_continuous | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"separated_space",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
separation_quotient (α : Type*) [uniform_space α] | quotient (separation_setoid α) | def | uniform_space.separation_quotient | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"separation_quotient",
"uniform_space"
] | The maximal separated quotient of a uniform space `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_eq_mk {x y : α} : (⟦x⟧ : separation_quotient α) = ⟦y⟧ ↔ inseparable x y | quotient.eq'.trans separation_rel_iff_inseparable | lemma | uniform_space.separation_quotient.mk_eq_mk | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"inseparable",
"separation_quotient",
"separation_rel_iff_inseparable"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
lift [separated_space β] (f : α → β) : (separation_quotient α → β) | if h : uniform_continuous f then
quotient.lift f (λ x y, eq_of_separated_of_uniform_continuous h)
else
λ x, f (nonempty.some ⟨x.out⟩) | def | uniform_space.separation_quotient.lift | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"lift",
"nonempty.some",
"separated_space",
"separation_quotient",
"uniform_continuous"
] | Factoring functions to a separated space through the separation quotient. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
lift_mk [separated_space β] {f : α → β} (h : uniform_continuous f) (a : α) :
lift f ⟦a⟧ = f a | by rw [lift, dif_pos h]; refl | lemma | uniform_space.separation_quotient.lift_mk | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"lift",
"lift_mk",
"separated_space",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_lift [separated_space β] (f : α → β) : uniform_continuous (lift f) | begin
by_cases hf : uniform_continuous f,
{ rw [lift, dif_pos hf], exact uniform_continuous_quotient_lift hf },
{ rw [lift, dif_neg hf], exact uniform_continuous_of_const (assume a b, rfl) }
end | lemma | uniform_space.separation_quotient.uniform_continuous_lift | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"lift",
"separated_space",
"uniform_continuous",
"uniform_continuous_of_const"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : α → β) : separation_quotient α → separation_quotient β | lift (quotient.mk ∘ f) | def | uniform_space.separation_quotient.map | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"lift",
"separation_quotient"
] | The separation quotient functor acting on functions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
map_mk {f : α → β} (h : uniform_continuous f) (a : α) : map f ⟦a⟧ = ⟦f a⟧ | by rw [map, lift_mk (uniform_continuous_quotient_mk.comp h)] | lemma | uniform_space.separation_quotient.map_mk | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"lift_mk",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
uniform_continuous_map (f : α → β) : uniform_continuous (map f) | uniform_continuous_lift (quotient.mk ∘ f) | lemma | uniform_space.separation_quotient.uniform_continuous_map | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_unique {f : α → β} (hf : uniform_continuous f)
{g : separation_quotient α → separation_quotient β}
(comm : quotient.mk ∘ f = g ∘ quotient.mk) : map f = g | by ext ⟨a⟩;
calc map f ⟦a⟧ = ⟦f a⟧ : map_mk hf a
... = g ⟦a⟧ : congr_fun comm a | lemma | uniform_space.separation_quotient.map_unique | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"comm",
"separation_quotient",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id : map (@id α) = id | map_unique uniform_continuous_id rfl | lemma | uniform_space.separation_quotient.map_id | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"map_id",
"uniform_continuous_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp {f : α → β} {g : β → γ} (hf : uniform_continuous f) (hg : uniform_continuous g) :
map g ∘ map f = map (g ∘ f) | (map_unique (hg.comp hf) $ by simp only [(∘), map_mk, hf, hg]).symm | lemma | uniform_space.separation_quotient.map_comp | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"map_comp",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
separation_prod {a₁ a₂ : α} {b₁ b₂ : β} : (a₁, b₁) ≈ (a₂, b₂) ↔ a₁ ≈ a₂ ∧ b₁ ≈ b₂ | begin
split,
{ assume h,
exact ⟨separated_of_uniform_continuous uniform_continuous_fst h,
separated_of_uniform_continuous uniform_continuous_snd h⟩ },
{ rintros ⟨eqv_α, eqv_β⟩ r r_in,
rw uniformity_prod at r_in,
rcases r_in with ⟨t_α, ⟨r_α, r_α_in, h_α⟩, t_β, ⟨r_β, r_β_in, h_β⟩, rfl⟩,
l... | lemma | uniform_space.separation_prod | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"uniform_continuous_fst",
"uniform_continuous_snd",
"uniformity_prod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
separated.prod [separated_space α] [separated_space β] : separated_space (α × β) | separated_def.2 $ assume x y H, prod.ext
(eq_of_separated_of_uniform_continuous uniform_continuous_fst H)
(eq_of_separated_of_uniform_continuous uniform_continuous_snd H) | instance | uniform_space.separated.prod | topology.uniform_space | src/topology/uniform_space/separation.lean | [
"tactic.apply_fun",
"topology.uniform_space.basic",
"topology.separation"
] | [
"prod.ext",
"separated_space",
"uniform_continuous_fst",
"uniform_continuous_snd"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on_filter (F : ι → α → β) (f : α → β) (p : filter ι) (p' : filter α) | ∀ u ∈ 𝓤 β, ∀ᶠ (n : ι × α) in (p ×ᶠ p'), (f n.snd, F n.fst n.snd) ∈ u | def | tendsto_uniformly_on_filter | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter"
] | A sequence of functions `Fₙ` converges uniformly on a filter `p'` to a limiting function `f`
with respect to the filter `p` if, for any entourage of the diagonal `u`, one has
`p ×ᶠ p'`-eventually `(f x, Fₙ x) ∈ u`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_on_filter_iff_tendsto :
tendsto_uniformly_on_filter F f p p' ↔
tendsto (λ q : ι × α, (f q.2, F q.1 q.2)) (p ×ᶠ p') (𝓤 β) | forall₂_congr $ λ u u_in, by simp [mem_map, filter.eventually, mem_prod_iff, preimage] | lemma | tendsto_uniformly_on_filter_iff_tendsto | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter.eventually",
"forall₂_congr",
"mem_map",
"tendsto_uniformly_on_filter"
] | A sequence of functions `Fₙ` converges uniformly on a filter `p'` to a limiting function `f` w.r.t.
filter `p` iff the function `(n, x) ↦ (f x, Fₙ x)` converges along `p ×ᶠ p'` to the uniformity.
In other words: one knows nothing about the behavior of `x` in this limit besides it being in `p'`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_on (F : ι → α → β) (f : α → β) (p : filter ι) (s : set α) | ∀ u ∈ 𝓤 β, ∀ᶠ n in p, ∀ (x : α), x ∈ s → (f x, F n x) ∈ u | def | tendsto_uniformly_on | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter"
] | A sequence of functions `Fₙ` converges uniformly on a set `s` to a limiting function `f` with
respect to the filter `p` if, for any entourage of the diagonal `u`, one has `p`-eventually
`(f x, Fₙ x) ∈ u` for all `x ∈ s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_on_iff_tendsto_uniformly_on_filter :
tendsto_uniformly_on F f p s ↔ tendsto_uniformly_on_filter F f p (𝓟 s) | begin
simp only [tendsto_uniformly_on, tendsto_uniformly_on_filter],
apply forall₂_congr,
simp_rw [eventually_prod_principal_iff],
simp,
end | lemma | tendsto_uniformly_on_iff_tendsto_uniformly_on_filter | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"forall₂_congr",
"tendsto_uniformly_on",
"tendsto_uniformly_on_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on_iff_tendsto {F : ι → α → β} {f : α → β} {p : filter ι} {s : set α} :
tendsto_uniformly_on F f p s ↔ tendsto (λ q : ι × α, (f q.2, F q.1 q.2)) (p ×ᶠ 𝓟 s) (𝓤 β) | by simp [tendsto_uniformly_on_iff_tendsto_uniformly_on_filter,
tendsto_uniformly_on_filter_iff_tendsto] | lemma | tendsto_uniformly_on_iff_tendsto | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter",
"tendsto_uniformly_on",
"tendsto_uniformly_on_filter_iff_tendsto",
"tendsto_uniformly_on_iff_tendsto_uniformly_on_filter"
] | A sequence of functions `Fₙ` converges uniformly on a set `s` to a limiting function `f` w.r.t.
filter `p` iff the function `(n, x) ↦ (f x, Fₙ x)` converges along `p ×ᶠ 𝓟 s` to the uniformity.
In other words: one knows nothing about the behavior of `x` in this limit besides it being in `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly (F : ι → α → β) (f : α → β) (p : filter ι) | ∀ u ∈ 𝓤 β, ∀ᶠ n in p, ∀ (x : α), (f x, F n x) ∈ u | def | tendsto_uniformly | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter"
] | A sequence of functions `Fₙ` converges uniformly to a limiting function `f` with respect to a
filter `p` if, for any entourage of the diagonal `u`, one has `p`-eventually
`(f x, Fₙ x) ∈ u` for all `x`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_iff_tendsto_uniformly_on_filter :
tendsto_uniformly F f p ↔ tendsto_uniformly_on_filter F f p ⊤ | begin
simp only [tendsto_uniformly, tendsto_uniformly_on_filter],
apply forall₂_congr,
simp_rw [← principal_univ, eventually_prod_principal_iff],
simp,
end | lemma | tendsto_uniformly_iff_tendsto_uniformly_on_filter | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"forall₂_congr",
"tendsto_uniformly",
"tendsto_uniformly_on_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly.tendsto_uniformly_on_filter
(h : tendsto_uniformly F f p) : tendsto_uniformly_on_filter F f p ⊤ | by rwa ← tendsto_uniformly_iff_tendsto_uniformly_on_filter | lemma | tendsto_uniformly.tendsto_uniformly_on_filter | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_uniformly",
"tendsto_uniformly_iff_tendsto_uniformly_on_filter",
"tendsto_uniformly_on_filter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_on_iff_tendsto_uniformly_comp_coe :
tendsto_uniformly_on F f p s ↔ tendsto_uniformly (λ i (x : s), F i x) (f ∘ coe) p | begin
apply forall₂_congr,
intros u hu,
simp,
end | lemma | tendsto_uniformly_on_iff_tendsto_uniformly_comp_coe | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"forall₂_congr",
"tendsto_uniformly",
"tendsto_uniformly_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
tendsto_uniformly_iff_tendsto {F : ι → α → β} {f : α → β} {p : filter ι} :
tendsto_uniformly F f p ↔ tendsto (λ q : ι × α, (f q.2, F q.1 q.2)) (p ×ᶠ ⊤) (𝓤 β) | by simp [tendsto_uniformly_iff_tendsto_uniformly_on_filter, tendsto_uniformly_on_filter_iff_tendsto] | lemma | tendsto_uniformly_iff_tendsto | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"filter",
"tendsto_uniformly",
"tendsto_uniformly_iff_tendsto_uniformly_on_filter",
"tendsto_uniformly_on_filter_iff_tendsto"
] | A sequence of functions `Fₙ` converges uniformly to a limiting function `f` w.r.t.
filter `p` iff the function `(n, x) ↦ (f x, Fₙ x)` converges along `p ×ᶠ ⊤` to the uniformity.
In other words: one knows nothing about the behavior of `x` in this limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_on_filter.tendsto_at (h : tendsto_uniformly_on_filter F f p p')
(hx : 𝓟 {x} ≤ p') : tendsto (λ n, F n x) p $ 𝓝 (f x) | begin
refine uniform.tendsto_nhds_right.mpr (λ u hu, mem_map.mpr _),
filter_upwards [(h u hu).curry],
intros i h,
simpa using (h.filter_mono hx),
end | lemma | tendsto_uniformly_on_filter.tendsto_at | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_uniformly_on_filter"
] | Uniform converence implies pointwise convergence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_on.tendsto_at (h : tendsto_uniformly_on F f p s) {x : α} (hx : x ∈ s) :
tendsto (λ n, F n x) p $ 𝓝 (f x) | h.tendsto_uniformly_on_filter.tendsto_at
(le_principal_iff.mpr $ mem_principal.mpr $ singleton_subset_iff.mpr $ hx) | lemma | tendsto_uniformly_on.tendsto_at | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_uniformly_on"
] | Uniform converence implies pointwise convergence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly.tendsto_at (h : tendsto_uniformly F f p) (x : α) :
tendsto (λ n, F n x) p $ 𝓝 (f x) | h.tendsto_uniformly_on_filter.tendsto_at le_top | lemma | tendsto_uniformly.tendsto_at | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"le_top",
"tendsto_uniformly"
] | Uniform converence implies pointwise convergence. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
tendsto_uniformly_on_univ :
tendsto_uniformly_on F f p univ ↔ tendsto_uniformly F f p | by simp [tendsto_uniformly_on, tendsto_uniformly] | lemma | tendsto_uniformly_on_univ | topology.uniform_space | src/topology/uniform_space/uniform_convergence.lean | [
"topology.separation",
"topology.uniform_space.basic",
"topology.uniform_space.cauchy"
] | [
"tendsto_uniformly",
"tendsto_uniformly_on"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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