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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|---|---|---|---|---|---|---|---|---|---|---|
is_bounded.all (s : set α) : is_bounded s | bounded_space.bounded_univ.subset s.subset_univ | lemma | bornology.is_bounded.all | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_cobounded.all (s : set α) : is_cobounded s | compl_compl s ▸ is_bounded.all sᶜ | lemma | bornology.is_cobounded.all | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [
"compl_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cobounded_eq_bot : cobounded α = ⊥ | cobounded_eq_bot_iff.2 ‹_› | lemma | bornology.cobounded_eq_bot | topology.bornology | src/topology/bornology/basic.lean | [
"order.filter.cofinite"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bornology.induced {α β : Type*} [bornology β] (f : α → β) : bornology α | { cobounded := comap f (cobounded β),
le_cofinite := (comap_mono (bornology.le_cofinite β)).trans (comap_cofinite_le _) } | def | bornology.induced | topology.bornology | src/topology/bornology/constructions.lean | [
"topology.bornology.basic"
] | [
"bornology"
] | Inverse image of a bornology. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cobounded_prod : cobounded (α × β) = (cobounded α).coprod (cobounded β) | rfl | lemma | bornology.cobounded_prod | topology.bornology | src/topology/bornology/constructions.lean | [
"topology.bornology.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_image_fst_and_snd {s : set (α × β)} :
is_bounded (prod.fst '' s) ∧ is_bounded (prod.snd '' s) ↔ is_bounded s | compl_mem_coprod.symm | lemma | bornology.is_bounded_image_fst_and_snd | topology.bornology | src/topology/bornology/constructions.lean | [
"topology.bornology.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded.fst_of_prod (h : is_bounded (s ×ˢ t)) (ht : t.nonempty) :
is_bounded s | fst_image_prod s ht ▸ (is_bounded_image_fst_and_snd.2 h).1 | lemma | bornology.is_bounded.fst_of_prod | topology.bornology | src/topology/bornology/constructions.lean | [
"topology.bornology.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded.snd_of_prod (h : is_bounded (s ×ˢ t)) (hs : s.nonempty) :
is_bounded t | snd_image_prod hs t ▸ (is_bounded_image_fst_and_snd.2 h).2 | lemma | bornology.is_bounded.snd_of_prod | topology.bornology | src/topology/bornology/constructions.lean | [
"topology.bornology.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded.prod (hs : is_bounded s) (ht : is_bounded t) : is_bounded (s ×ˢ t) | is_bounded_image_fst_and_snd.1
⟨hs.subset $ fst_image_prod_subset _ _, ht.subset $ snd_image_prod_subset _ _⟩ | lemma | bornology.is_bounded.prod | topology.bornology | src/topology/bornology/constructions.lean | [
"topology.bornology.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_prod_of_nonempty (hne : set.nonempty (s ×ˢ t)) :
is_bounded (s ×ˢ t) ↔ is_bounded s ∧ is_bounded t | ⟨λ h, ⟨h.fst_of_prod hne.snd, h.snd_of_prod hne.fst⟩, λ h, h.1.prod h.2⟩ | lemma | bornology.is_bounded_prod_of_nonempty | topology.bornology | src/topology/bornology/constructions.lean | [
"topology.bornology.basic"
] | [
"set.nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_prod : is_bounded (s ×ˢ t) ↔ s = ∅ ∨ t = ∅ ∨ is_bounded s ∧ is_bounded t | begin
rcases s.eq_empty_or_nonempty with rfl|hs, { simp },
rcases t.eq_empty_or_nonempty with rfl|ht, { simp },
simp only [hs.ne_empty, ht.ne_empty, is_bounded_prod_of_nonempty (hs.prod ht), false_or]
end | lemma | bornology.is_bounded_prod | topology.bornology | src/topology/bornology/constructions.lean | [
"topology.bornology.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_prod_self : is_bounded (s ×ˢ s) ↔ is_bounded s | begin
rcases s.eq_empty_or_nonempty with rfl|hs, { simp },
exact (is_bounded_prod_of_nonempty (hs.prod hs)).trans (and_self _)
end | lemma | bornology.is_bounded_prod_self | topology.bornology | src/topology/bornology/constructions.lean | [
"topology.bornology.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cobounded_pi : cobounded (Π i, π i) = filter.Coprod (λ i, cobounded (π i)) | rfl | lemma | bornology.cobounded_pi | topology.bornology | src/topology/bornology/constructions.lean | [
"topology.bornology.basic"
] | [
"filter.Coprod"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forall_is_bounded_image_eval_iff {s : set (Π i, π i)} :
(∀ i, is_bounded (eval i '' s)) ↔ is_bounded s | compl_mem_Coprod.symm | lemma | bornology.forall_is_bounded_image_eval_iff | topology.bornology | src/topology/bornology/constructions.lean | [
"topology.bornology.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded.pi (h : ∀ i, is_bounded (S i)) : is_bounded (pi univ S) | forall_is_bounded_image_eval_iff.1 $ λ i, (h i).subset eval_image_univ_pi_subset | lemma | bornology.is_bounded.pi | topology.bornology | src/topology/bornology/constructions.lean | [
"topology.bornology.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_pi_of_nonempty (hne : (pi univ S).nonempty) :
is_bounded (pi univ S) ↔ ∀ i, is_bounded (S i) | ⟨λ H i, @eval_image_univ_pi _ _ _ i hne ▸ forall_is_bounded_image_eval_iff.2 H i, is_bounded.pi⟩ | lemma | bornology.is_bounded_pi_of_nonempty | topology.bornology | src/topology/bornology/constructions.lean | [
"topology.bornology.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_pi : is_bounded (pi univ S) ↔ (∃ i, S i = ∅) ∨ ∀ i, is_bounded (S i) | begin
by_cases hne : ∃ i, S i = ∅,
{ simp [hne, univ_pi_eq_empty_iff.2 hne] },
{ simp only [hne, false_or],
simp only [not_exists, ← ne.def, ←nonempty_iff_ne_empty, ← univ_pi_nonempty_iff] at hne,
exact is_bounded_pi_of_nonempty hne }
end | lemma | bornology.is_bounded_pi | topology.bornology | src/topology/bornology/constructions.lean | [
"topology.bornology.basic"
] | [
"not_exists"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_induced {α β : Type*} [bornology β] {f : α → β} {s : set α} :
@is_bounded α (bornology.induced f) s ↔ is_bounded (f '' s) | compl_mem_comap | lemma | bornology.is_bounded_induced | topology.bornology | src/topology/bornology/constructions.lean | [
"topology.bornology.basic"
] | [
"bornology",
"bornology.induced"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_bounded_image_subtype_coe {p : α → Prop} {s : set {x // p x}} :
is_bounded (coe '' s : set α) ↔ is_bounded s | is_bounded_induced.symm | lemma | bornology.is_bounded_image_subtype_coe | topology.bornology | src/topology/bornology/constructions.lean | [
"topology.bornology.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_space_induced_iff {α β : Type*} [bornology β] {f : α → β} :
@bounded_space α (bornology.induced f) ↔ is_bounded (range f) | by rw [← is_bounded_univ, is_bounded_induced, image_univ] | lemma | bounded_space_induced_iff | topology.bornology | src/topology/bornology/constructions.lean | [
"topology.bornology.basic"
] | [
"bornology",
"bornology.induced",
"bounded_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_space_subtype_iff {p : α → Prop} : bounded_space (subtype p) ↔ is_bounded {x | p x} | by rw [bounded_space_induced_iff, subtype.range_coe_subtype] | lemma | bounded_space_subtype_iff | topology.bornology | src/topology/bornology/constructions.lean | [
"topology.bornology.basic"
] | [
"bounded_space",
"bounded_space_induced_iff",
"subtype.range_coe_subtype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
bounded_space_coe_set_iff {s : set α} : bounded_space s ↔ is_bounded s | bounded_space_subtype_iff | lemma | bounded_space_coe_set_iff | topology.bornology | src/topology/bornology/constructions.lean | [
"topology.bornology.basic"
] | [
"bounded_space",
"bounded_space_subtype_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_bounded_map (α β : Type*) [bornology α] [bornology β] | (to_fun : α → β)
(comap_cobounded_le' : (cobounded β).comap to_fun ≤ cobounded α) | structure | locally_bounded_map | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [
"bornology"
] | The type of bounded maps from `α` to `β`, the maps which send a bounded set to a bounded set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
locally_bounded_map_class (F : Type*) (α β : out_param $ Type*) [bornology α]
[bornology β]
extends fun_like F α (λ _, β) | (comap_cobounded_le (f : F) : (cobounded β).comap f ≤ cobounded α) | class | locally_bounded_map_class | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [
"bornology",
"fun_like"
] | `locally_bounded_map_class F α β` states that `F` is a type of bounded maps.
You should extend this class when you extend `locally_bounded_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_bounded.image [bornology α] [bornology β] [locally_bounded_map_class F α β] {f : F}
{s : set α} (hs : is_bounded s) : is_bounded (f '' s) | comap_cobounded_le_iff.1 (comap_cobounded_le f) hs | lemma | is_bounded.image | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [
"bornology",
"locally_bounded_map_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe {f : locally_bounded_map α β} : f.to_fun = (f : α → β) | rfl | lemma | locally_bounded_map.to_fun_eq_coe | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [
"locally_bounded_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : locally_bounded_map α β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | locally_bounded_map.ext | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [
"fun_like.ext",
"locally_bounded_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : locally_bounded_map α β) (f' : α → β) (h : f' = f) :
locally_bounded_map α β | ⟨f', h.symm ▸ f.comap_cobounded_le'⟩ | def | locally_bounded_map.copy | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [
"locally_bounded_map"
] | Copy of a `locally_bounded_map` with a new `to_fun` equal to the old one. Useful to fix
definitional equalities. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_copy (f : locally_bounded_map α β) (f' : α → β) (h : f' = f) :
⇑(f.copy f' h) = f' | rfl | lemma | locally_bounded_map.coe_copy | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [
"locally_bounded_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : locally_bounded_map α β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | locally_bounded_map.copy_eq | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [
"fun_like.ext'",
"locally_bounded_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_map_bounded (f : α → β) (h) : locally_bounded_map α β | ⟨f, comap_cobounded_le_iff.2 h⟩ | def | locally_bounded_map.of_map_bounded | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [
"locally_bounded_map"
] | Construct a `locally_bounded_map` from the fact that the function maps bounded sets to bounded
sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of_map_bounded (f : α → β) {h} : ⇑(of_map_bounded f h) = f | rfl | lemma | locally_bounded_map.coe_of_map_bounded | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
of_map_bounded_apply (f : α → β) {h} (a : α) : of_map_bounded f h a = f a | rfl | lemma | locally_bounded_map.of_map_bounded_apply | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : locally_bounded_map α α | ⟨id, comap_id.le⟩ | def | locally_bounded_map.id | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [
"locally_bounded_map"
] | `id` as a `locally_bounded_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(locally_bounded_map.id α) = id | rfl | lemma | locally_bounded_map.coe_id | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [
"locally_bounded_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (a : α) : locally_bounded_map.id α a = a | rfl | lemma | locally_bounded_map.id_apply | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [
"locally_bounded_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (f : locally_bounded_map β γ) (g : locally_bounded_map α β) : locally_bounded_map α γ | { to_fun := f ∘ g,
comap_cobounded_le' :=
comap_comap.ge.trans $ (comap_mono f.comap_cobounded_le').trans g.comap_cobounded_le' } | def | locally_bounded_map.comp | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [
"locally_bounded_map"
] | Composition of `locally_bounded_map`s as a `locally_bounded_map`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (f : locally_bounded_map β γ) (g : locally_bounded_map α β) :
⇑(f.comp g) = f ∘ g | rfl | lemma | locally_bounded_map.coe_comp | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [
"locally_bounded_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : locally_bounded_map β γ) (g : locally_bounded_map α β) (a : α) :
f.comp g a = f (g a) | rfl | lemma | locally_bounded_map.comp_apply | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [
"locally_bounded_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : locally_bounded_map γ δ) (g : locally_bounded_map β γ)
(h : locally_bounded_map α β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | locally_bounded_map.comp_assoc | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [
"locally_bounded_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : locally_bounded_map α β) :
f.comp (locally_bounded_map.id α) = f | ext $ λ a, rfl | lemma | locally_bounded_map.comp_id | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [
"locally_bounded_map",
"locally_bounded_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : locally_bounded_map α β) :
(locally_bounded_map.id β).comp f = f | ext $ λ a, rfl | lemma | locally_bounded_map.id_comp | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [
"locally_bounded_map",
"locally_bounded_map.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : locally_bounded_map β γ} {f : locally_bounded_map α β}
(hf : surjective f) :
g₁.comp f = g₂.comp f ↔ g₁ = g₂ | ⟨λ h, ext $ hf.forall.2 $ fun_like.ext_iff.1 h, congr_arg _⟩ | lemma | locally_bounded_map.cancel_right | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [
"locally_bounded_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g : locally_bounded_map β γ} {f₁ f₂ : locally_bounded_map α β}
(hg : injective g) :
g.comp f₁ = g.comp f₂ ↔ f₁ = f₂ | ⟨λ h, ext $ λ a, hg $ by rw [←comp_apply, h, comp_apply], congr_arg _⟩ | lemma | locally_bounded_map.cancel_left | topology.bornology | src/topology/bornology/hom.lean | [
"topology.bornology.basic"
] | [
"locally_bounded_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Born | bundled bornology | def | Born | topology.category | src/topology/category/Born.lean | [
"category_theory.concrete_category.bundled_hom",
"topology.bornology.hom"
] | [
"bornology"
] | The category of bornologies. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (α : Type*) [bornology α] : Born | bundled.of α | def | Born.of | topology.category | src/topology/category/Born.lean | [
"category_theory.concrete_category.bundled_hom",
"topology.bornology.hom"
] | [
"Born",
"bornology"
] | Construct a bundled `Born` from a `bornology`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Compactum | monad.algebra β | def | Compactum | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [] | The type `Compactum` of Compacta, defined as algebras for the ultrafilter monad. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget : Compactum ⥤ Type* | monad.forget _ | def | Compactum.forget | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum"
] | The forgetful functor to Type* | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
free : Type* ⥤ Compactum | monad.free _ | def | Compactum.free | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum",
"free"
] | The "free" Compactum functor. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
adj : free ⊣ forget | monad.adj _ | def | Compactum.adj | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"adj",
"free"
] | The adjunction between `free` and `forget`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
str (X : Compactum) : ultrafilter X → X | X.a | def | Compactum.str | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum",
"ultrafilter"
] | The structure map for a compactum, essentially sending an ultrafilter to its limit. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
join (X : Compactum) : ultrafilter (ultrafilter X) → ultrafilter X | β .μ.app _ | def | Compactum.join | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum",
"ultrafilter"
] | The monadic join. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
incl (X : Compactum) : X → ultrafilter X | β .η.app _ | def | Compactum.incl | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum",
"ultrafilter"
] | The inclusion of `X` into `ultrafilter X`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
str_incl (X : Compactum) (x : X) : X.str (X.incl x) = x | begin
change (β .η.app _ ≫ X.a) _ = _,
rw monad.algebra.unit,
refl,
end | lemma | Compactum.str_incl | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
str_hom_commute (X Y : Compactum) (f : X ⟶ Y) (xs : ultrafilter X) :
f (X.str xs) = Y.str (map f xs) | begin
change (X.a ≫ f.f) _ = _,
rw ←f.h,
refl,
end | lemma | Compactum.str_hom_commute | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
join_distrib (X : Compactum) (uux : ultrafilter (ultrafilter X)) :
X.str (X.join uux) = X.str (map X.str uux) | begin
change (β .μ.app _ ≫ X.a) _ = _,
rw monad.algebra.assoc,
refl,
end | lemma | Compactum.join_distrib | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_iff {X : Compactum} (S : set X) : is_closed S ↔
(∀ F : ultrafilter X, S ∈ F → X.str F ∈ S) | begin
rw ← is_open_compl_iff,
split,
{ intros cond F h,
by_contradiction c,
specialize cond F c,
rw compl_mem_iff_not_mem at cond,
contradiction },
{ intros h1 F h2,
specialize h1 F,
cases F.mem_or_compl_mem S, exacts [absurd (h1 h) h2, h] }
end | theorem | Compactum.is_closed_iff | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum",
"by_contradiction",
"is_closed",
"is_open_compl_iff",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
basic {X : Compactum} (A : set X) : set (ultrafilter X) | {F | A ∈ F} | def | Compactum.basic | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum",
"ultrafilter"
] | A local definition used only in the proofs. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cl {X : Compactum} (A : set X) : set X | X.str '' (basic A) | def | Compactum.cl | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum"
] | A local definition used only in the proofs. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
basic_inter {X : Compactum} (A B : set X) : basic (A ∩ B) = basic A ∩ basic B | begin
ext G,
split,
{ intro hG,
split; filter_upwards [hG] with _,
exacts [and.left, and.right] },
{ rintros ⟨h1, h2⟩,
exact inter_mem h1 h2 }
end | lemma | Compactum.basic_inter | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_cl {X : Compactum} (A : set X) : A ⊆ cl A | λ a ha, ⟨X.incl a, ha,by simp⟩ | lemma | Compactum.subset_cl | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cl_cl {X : Compactum} (A : set X) : cl (cl A) ⊆ cl A | begin
rintros _ ⟨F,hF,rfl⟩,
-- Notation to be used in this proof.
let fsu := finset (set (ultrafilter X)),
let ssu := set (set (ultrafilter X)),
let ι : fsu → ssu := coe,
let C0 : ssu := {Z | ∃ B ∈ F, X.str ⁻¹' B = Z},
let AA := {G : ultrafilter X | A ∈ G},
let C1 := insert AA C0,
let C2 := finite_int... | theorem | Compactum.cl_cl | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum",
"filter.nonempty_of_mem",
"finset",
"has_finite_inter",
"set.nonempty",
"set.preimage_inter",
"subtype.val_eq_coe",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_cl {X : Compactum} (A : set X) : is_closed (cl A) | begin
rw is_closed_iff,
intros F hF,
exact cl_cl _ ⟨F, hF, rfl⟩,
end | lemma | Compactum.is_closed_cl | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
str_eq_of_le_nhds {X : Compactum} (F : ultrafilter X) (x : X) :
↑F ≤ 𝓝 x → X.str F = x | begin
-- Notation to be used in this proof.
let fsu := finset (set (ultrafilter X)),
let ssu := set (set (ultrafilter X)),
let ι : fsu → ssu := coe,
let T0 : ssu := { S | ∃ A ∈ F, S = basic A },
let AA := (X.str ⁻¹' {x}),
let T1 := insert AA T0,
let T2 := finite_inter_closure T1,
intro cond,
-- If F... | lemma | Compactum.str_eq_of_le_nhds | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum",
"by_contradiction",
"finset",
"is_closed",
"le_nhds_iff",
"set.nonempty",
"ultrafilter",
"ultrafilter.compl_mem_iff_not_mem",
"ultrafilter.exists_ultrafilter_of_finite_inter_nonempty",
"ultrafilter.mem_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
le_nhds_of_str_eq {X : Compactum} (F : ultrafilter X) (x : X) :
X.str F = x → ↑F ≤ 𝓝 x | λ h, le_nhds_iff.mpr (λ s hx hs, hs _ $ by rwa h) | lemma | Compactum.le_nhds_of_str_eq | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum",
"ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Lim_eq_str {X : Compactum} (F : ultrafilter X) : F.Lim = X.str F | begin
rw [ultrafilter.Lim_eq_iff_le_nhds, le_nhds_iff],
tauto,
end | lemma | Compactum.Lim_eq_str | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum",
"le_nhds_iff",
"ultrafilter",
"ultrafilter.Lim_eq_iff_le_nhds"
] | The structure map of a compactum actually computes limits. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
cl_eq_closure {X : Compactum} (A : set X) : cl A = closure A | begin
ext,
rw mem_closure_iff_ultrafilter,
split,
{ rintro ⟨F, h1, h2⟩,
exact ⟨F, h1, le_nhds_of_str_eq _ _ h2⟩ },
{ rintro ⟨F, h1, h2⟩,
exact ⟨F, h1, str_eq_of_le_nhds _ _ h2⟩ }
end | lemma | Compactum.cl_eq_closure | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum",
"closure",
"mem_closure_iff_ultrafilter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous_of_hom {X Y : Compactum} (f : X ⟶ Y) : continuous f | begin
rw continuous_iff_ultrafilter,
intros x _ h,
rw [tendsto, ← coe_map],
apply le_nhds_of_str_eq,
rw [← str_hom_commute, str_eq_of_le_nhds _ x h]
end | lemma | Compactum.continuous_of_hom | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum",
"continuous",
"continuous_iff_ultrafilter"
] | Any morphism of compacta is continuous. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of_topological_space (X : Type*) [topological_space X]
[compact_space X] [t2_space X] : Compactum | { A := X,
a := ultrafilter.Lim,
unit' := by {ext x, exact Lim_eq (pure_le_nhds _) },
assoc' := begin
ext FF,
change ultrafilter (ultrafilter X) at FF,
set x := (ultrafilter.map ultrafilter.Lim FF).Lim with c1,
have c2 : ∀ (U : set X) (F : ultrafilter X), F.Lim ∈ U → is_open U → U ∈ F,
{ intros... | def | Compactum.of_topological_space | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum",
"Lim",
"Lim_eq",
"compact_space",
"is_open",
"is_open.mem_nhds",
"le_nhds_iff",
"pure_le_nhds",
"t2_space",
"topological_space",
"ultrafilter",
"ultrafilter.Lim",
"ultrafilter.le_nhds_Lim",
"ultrafilter.map"
] | Given any compact Hausdorff space, we construct a Compactum. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
hom_of_continuous {X Y : Compactum} (f : X → Y) (cont : continuous f) : X ⟶ Y | { f := f,
h' := begin
rw continuous_iff_ultrafilter at cont,
ext (F : ultrafilter X),
specialize cont (X.str F) F (le_nhds_of_str_eq F (X.str F) rfl),
have := str_eq_of_le_nhds (ultrafilter.map f F) _ cont,
simpa only [←this, types_comp_apply, of_type_functor_map],
end } | def | Compactum.hom_of_continuous | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum",
"cont",
"continuous",
"continuous_iff_ultrafilter",
"ultrafilter",
"ultrafilter.map"
] | Any continuous map between Compacta is a morphism of compacta. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Compactum_to_CompHaus : Compactum ⥤ CompHaus | { obj := λ X, { to_Top := { α := X } },
map := λ X Y f,
{ to_fun := f,
continuous_to_fun := Compactum.continuous_of_hom _ }} | def | Compactum_to_CompHaus | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"CompHaus",
"Compactum",
"Compactum.continuous_of_hom"
] | The functor functor from Compactum to CompHaus. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
full : full Compactum_to_CompHaus.{u} | { preimage := λ X Y f, Compactum.hom_of_continuous f.1 f.2 } | def | Compactum_to_CompHaus.full | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum.hom_of_continuous"
] | The functor Compactum_to_CompHaus is full. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
faithful : faithful Compactum_to_CompHaus | {} | lemma | Compactum_to_CompHaus.faithful | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum_to_CompHaus"
] | The functor Compactum_to_CompHaus is faithful. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
iso_of_topological_space {D : CompHaus} :
Compactum_to_CompHaus.obj (Compactum.of_topological_space D) ≅ D | { hom :=
{ to_fun := id,
continuous_to_fun := continuous_def.2 $ λ _ h, by {rw is_open_iff_ultrafilter' at h, exact h} },
inv :=
{ to_fun := id,
continuous_to_fun := continuous_def.2 $
λ _ h1, by {rw is_open_iff_ultrafilter', intros _ h2, exact h1 _ h2} } } | def | Compactum_to_CompHaus.iso_of_topological_space | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"CompHaus",
"Compactum.of_topological_space",
"is_open_iff_ultrafilter'"
] | This definition is used to prove essential surjectivity of Compactum_to_CompHaus. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ess_surj : ess_surj Compactum_to_CompHaus | { mem_ess_image := λ X, ⟨Compactum.of_topological_space X, ⟨iso_of_topological_space⟩⟩ } | lemma | Compactum_to_CompHaus.ess_surj | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum_to_CompHaus"
] | The functor Compactum_to_CompHaus is essentially surjective. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_equivalence : is_equivalence Compactum_to_CompHaus | begin
apply equivalence.of_fully_faithfully_ess_surj _,
exact Compactum_to_CompHaus.full,
exact Compactum_to_CompHaus.faithful,
exact Compactum_to_CompHaus.ess_surj,
end | instance | Compactum_to_CompHaus.is_equivalence | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Compactum_to_CompHaus",
"Compactum_to_CompHaus.ess_surj",
"Compactum_to_CompHaus.faithful",
"Compactum_to_CompHaus.full"
] | The functor Compactum_to_CompHaus is an equivalence of categories. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Compactum_to_CompHaus_comp_forget :
Compactum_to_CompHaus ⋙ category_theory.forget CompHaus ≅ Compactum.forget | nat_iso.of_components (λ X, eq_to_iso rfl) $
by { intros X Y f, dsimp, simpa } | def | Compactum_to_CompHaus_comp_forget | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"CompHaus",
"Compactum.forget",
"Compactum_to_CompHaus",
"category_theory.forget"
] | The forgetful functors of `Compactum` and `CompHaus` are compatible via
`Compactum_to_CompHaus`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CompHaus.forget_creates_limits : creates_limits (forget CompHaus) | begin
let e : forget CompHaus ≅ Compactum_to_CompHaus.inv ⋙ Compactum.forget :=
_ ≪≫ iso_whisker_left _ Compactum_to_CompHaus_comp_forget,
swap,
refine _ ≪≫ functor.associator _ _ _,
refine (functor.left_unitor _).symm ≪≫ _,
refine iso_whisker_right _ _,
exact Compactum_to_CompHaus.as_equivalence.symm.u... | instance | CompHaus.forget_creates_limits | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"CompHaus",
"Compactum.forget",
"Compactum_to_CompHaus_comp_forget"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Profinite.forget_creates_limits : creates_limits (forget Profinite) | begin
change creates_limits (Profinite_to_CompHaus ⋙ forget _),
apply_instance,
end | instance | Profinite.forget_creates_limits | topology.category | src/topology/category/Compactum.lean | [
"category_theory.monad.types",
"category_theory.monad.limits",
"category_theory.equivalence",
"topology.category.CompHaus.basic",
"topology.category.Profinite.basic",
"data.set.constructions"
] | [
"Profinite",
"Profinite_to_CompHaus"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
Locale | Frmᵒᵖ | def | Locale | topology.category | src/topology/category/Locale.lean | [
"order.category.Frm"
] | [] | The category of locales. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (α : Type*) [frame α] : Locale | op $ Frm.of α | def | Locale.of | topology.category | src/topology/category/Locale.lean | [
"order.category.Frm"
] | [
"Frm.of",
"Locale"
] | Construct a bundled `Locale` from a `frame`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
Top_to_Locale : Top ⥤ Locale | Top_op_to_Frame.right_op | def | Top_to_Locale | topology.category | src/topology/category/Locale.lean | [
"order.category.Frm"
] | [
"Locale",
"Top"
] | The forgetful functor from `Top` to `Locale` which forgets that the space has "enough points". | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
CompHaus_to_Locale.faithful : faithful (CompHaus_to_Top ⋙ Top_to_Locale.{u}) | ⟨λ X Y f g h, by { dsimp at h, exact opens.comap_injective (quiver.hom.op_inj h) }⟩ | instance | CompHaus_to_Locale.faithful | topology.category | src/topology/category/Locale.lean | [
"order.category.Frm"
] | [
"CompHaus_to_Top",
"quiver.hom.op_inj"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
TopCommRing | (α : Type u)
[is_comm_ring : comm_ring α]
[is_topological_space : topological_space α]
[is_topological_ring : topological_ring α] | structure | TopCommRing | topology.category | src/topology/category/TopCommRing.lean | [
"algebra.category.Ring.basic",
"topology.category.Top.basic",
"topology.algebra.ring.basic"
] | [
"comm_ring",
"topological_ring",
"topological_space"
] | A bundled topological commutative ring. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (X : Type u) [comm_ring X] [topological_space X] [topological_ring X] : TopCommRing | ⟨X⟩ | def | TopCommRing.of | topology.category | src/topology/category/TopCommRing.lean | [
"algebra.category.Ring.basic",
"topology.category.Top.basic",
"topology.algebra.ring.basic"
] | [
"TopCommRing",
"comm_ring",
"topological_ring",
"topological_space"
] | Construct a bundled `TopCommRing` from the underlying type and the appropriate typeclasses. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of (X : Type u) [comm_ring X] [topological_space X] [topological_ring X] :
(of X : Type u) = X | rfl | lemma | TopCommRing.coe_of | topology.category | src/topology/category/TopCommRing.lean | [
"algebra.category.Ring.basic",
"topology.category.Top.basic",
"topology.algebra.ring.basic"
] | [
"comm_ring",
"topological_ring",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_topological_space (R : TopCommRing) :
topological_space ((forget TopCommRing).obj R) | R.is_topological_space | instance | TopCommRing.forget_topological_space | topology.category | src/topology/category/TopCommRing.lean | [
"algebra.category.Ring.basic",
"topology.category.Top.basic",
"topology.algebra.ring.basic"
] | [
"TopCommRing",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_comm_ring (R : TopCommRing) :
comm_ring ((forget TopCommRing).obj R) | R.is_comm_ring | instance | TopCommRing.forget_comm_ring | topology.category | src/topology/category/TopCommRing.lean | [
"algebra.category.Ring.basic",
"topology.category.Top.basic",
"topology.algebra.ring.basic"
] | [
"TopCommRing",
"comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_topological_ring (R : TopCommRing) :
topological_ring ((forget TopCommRing).obj R) | R.is_topological_ring | instance | TopCommRing.forget_topological_ring | topology.category | src/topology/category/TopCommRing.lean | [
"algebra.category.Ring.basic",
"topology.category.Top.basic",
"topology.algebra.ring.basic"
] | [
"TopCommRing",
"topological_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget_to_CommRing : has_forget₂ TopCommRing CommRing | has_forget₂.mk'
(λ R, CommRing.of R)
(λ x, rfl)
(λ R S f, f.val)
(λ R S f, heq.rfl) | instance | TopCommRing.has_forget_to_CommRing | topology.category | src/topology/category/TopCommRing.lean | [
"algebra.category.Ring.basic",
"topology.category.Top.basic",
"topology.algebra.ring.basic"
] | [
"CommRing",
"CommRing.of",
"TopCommRing"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_to_CommRing_topological_space (R : TopCommRing) :
topological_space ((forget₂ TopCommRing CommRing).obj R) | R.is_topological_space | instance | TopCommRing.forget_to_CommRing_topological_space | topology.category | src/topology/category/TopCommRing.lean | [
"algebra.category.Ring.basic",
"topology.category.Top.basic",
"topology.algebra.ring.basic"
] | [
"CommRing",
"TopCommRing",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
has_forget_to_Top : has_forget₂ TopCommRing Top | has_forget₂.mk'
(λ R, Top.of R)
(λ x, rfl)
(λ R S f, ⟨⇑f.1, f.2⟩)
(λ R S f, heq.rfl) | instance | TopCommRing.has_forget_to_Top | topology.category | src/topology/category/TopCommRing.lean | [
"algebra.category.Ring.basic",
"topology.category.Top.basic",
"topology.algebra.ring.basic"
] | [
"Top",
"Top.of",
"TopCommRing"
] | The forgetful functor to Top. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
forget_to_Top_comm_ring (R : TopCommRing) :
comm_ring ((forget₂ TopCommRing Top).obj R) | R.is_comm_ring | instance | TopCommRing.forget_to_Top_comm_ring | topology.category | src/topology/category/TopCommRing.lean | [
"algebra.category.Ring.basic",
"topology.category.Top.basic",
"topology.algebra.ring.basic"
] | [
"Top",
"TopCommRing",
"comm_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
forget_to_Top_topological_ring (R : TopCommRing) :
topological_ring ((forget₂ TopCommRing Top).obj R) | R.is_topological_ring | instance | TopCommRing.forget_to_Top_topological_ring | topology.category | src/topology/category/TopCommRing.lean | [
"algebra.category.Ring.basic",
"topology.category.Top.basic",
"topology.algebra.ring.basic"
] | [
"Top",
"TopCommRing",
"topological_ring"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
UniformSpace : Type (u+1) | bundled uniform_space | def | UniformSpace | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"uniform_space"
] | A (bundled) uniform space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
of (α : Type u) [uniform_space α] : UniformSpace | ⟨α⟩ | def | UniformSpace.of | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"UniformSpace",
"uniform_space"
] | Construct a bundled `UniformSpace` from the underlying type and the typeclass. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_of (X : Type u) [uniform_space X] : (of X : Type u) = X | rfl | lemma | UniformSpace.coe_of | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"uniform_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp {X Y Z : UniformSpace} (f : X ⟶ Y) (g : Y ⟶ Z) :
(f ≫ g : X → Z) = g ∘ f | rfl | lemma | UniformSpace.coe_comp | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"UniformSpace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_id (X : UniformSpace) : (𝟙 X : X → X) = id | rfl | lemma | UniformSpace.coe_id | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"UniformSpace"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk {X Y : UniformSpace} (f : X → Y) (hf : uniform_continuous f) :
((⟨f, hf⟩ : X ⟶ Y) : X → Y) = f | rfl | lemma | UniformSpace.coe_mk | topology.category | src/topology/category/UniformSpace.lean | [
"category_theory.adjunction.reflective",
"category_theory.concrete_category.unbundled_hom",
"category_theory.monad.limits",
"topology.category.Top.basic",
"topology.uniform_space.completion"
] | [
"UniformSpace",
"uniform_continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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