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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|---|---|---|---|---|---|---|---|---|---|---|
coe_comp (f : β →Co γ) (g : α →Co β) : (f.comp g : α → γ) = f ∘ g | rfl | lemma | continuous_order_hom.coe_comp | topology.order.hom | src/topology/order/hom/basic.lean | [
"order.hom.basic",
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (f : β →Co γ) (g : α →Co β) (a : α) : (f.comp g) a = f (g a) | rfl | lemma | continuous_order_hom.comp_apply | topology.order.hom | src/topology/order/hom/basic.lean | [
"order.hom.basic",
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (f : γ →Co δ) (g : β →Co γ) (h : α →Co β) :
(f.comp g).comp h = f.comp (g.comp h) | rfl | lemma | continuous_order_hom.comp_assoc | topology.order.hom | src/topology/order/hom/basic.lean | [
"order.hom.basic",
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : α →Co β) : f.comp (continuous_order_hom.id α) = f | ext $ λ a, rfl | lemma | continuous_order_hom.comp_id | topology.order.hom | src/topology/order/hom/basic.lean | [
"order.hom.basic",
"topology.continuous_function.basic"
] | [
"continuous_order_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : α →Co β) : (continuous_order_hom.id β).comp f = f | ext $ λ a, rfl | lemma | continuous_order_hom.id_comp | topology.order.hom | src/topology/order/hom/basic.lean | [
"order.hom.basic",
"topology.continuous_function.basic"
] | [
"continuous_order_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : β →Co γ} {f : α →Co 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 | continuous_order_hom.cancel_right | topology.order.hom | src/topology/order/hom/basic.lean | [
"order.hom.basic",
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g : β →Co γ} {f₁ f₂ : α →Co 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 | continuous_order_hom.cancel_left | topology.order.hom | src/topology/order/hom/basic.lean | [
"order.hom.basic",
"topology.continuous_function.basic"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pseudo_epimorphism (α β : Type*) [preorder α] [preorder β] extends α →o β | (exists_map_eq_of_map_le' ⦃a : α⦄ ⦃b : β⦄ : to_fun a ≤ b → ∃ c, a ≤ c ∧ to_fun c = b) | structure | pseudo_epimorphism | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [] | The type of pseudo-epimorphisms, aka p-morphisms, aka bounded maps, from `α` to `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
esakia_hom (α β : Type*) [topological_space α] [preorder α] [topological_space β]
[preorder β] extends α →Co β | (exists_map_eq_of_map_le' ⦃a : α⦄ ⦃b : β⦄ : to_fun a ≤ b → ∃ c, a ≤ c ∧ to_fun c = b) | structure | esakia_hom | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"topological_space"
] | The type of Esakia morphisms, aka continuous pseudo-epimorphisms, from `α` to `β`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_epimorphism_class (F : Type*) (α β : out_param $ Type*) [preorder α] [preorder β]
extends rel_hom_class F ((≤) : α → α → Prop) ((≤) : β → β → Prop) | (exists_map_eq_of_map_le (f : F) ⦃a : α⦄ ⦃b : β⦄ : f a ≤ b → ∃ c, a ≤ c ∧ f c = b) | class | pseudo_epimorphism_class | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"rel_hom_class"
] | `pseudo_epimorphism_class F α β` states that `F` is a type of `⊔`-preserving morphisms.
You should extend this class when you extend `pseudo_epimorphism`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
esakia_hom_class (F : Type*) (α β : out_param $ Type*) [topological_space α] [preorder α]
[topological_space β] [preorder β]
extends continuous_order_hom_class F α β | (exists_map_eq_of_map_le (f : F) ⦃a : α⦄ ⦃b : β⦄ : f a ≤ b → ∃ c, a ≤ c ∧ f c = b) | class | esakia_hom_class | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"continuous_order_hom_class",
"topological_space"
] | `esakia_hom_class F α β` states that `F` is a type of lattice morphisms.
You should extend this class when you extend `esakia_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
pseudo_epimorphism_class.to_top_hom_class [partial_order α] [order_top α] [preorder β]
[order_top β] [pseudo_epimorphism_class F α β] : top_hom_class F α β | { map_top := λ f, let ⟨b, h⟩ := exists_map_eq_of_map_le f (@le_top _ _ _ $ f ⊤) in
by rw [←top_le_iff.1 h.1, h.2]
.. ‹pseudo_epimorphism_class F α β› } | instance | pseudo_epimorphism_class.to_top_hom_class | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"le_top",
"order_top",
"pseudo_epimorphism_class",
"top_hom_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
order_iso_class.to_pseudo_epimorphism_class [preorder α] [preorder β]
[order_iso_class F α β] : pseudo_epimorphism_class F α β | { exists_map_eq_of_map_le :=
λ f a b h, ⟨equiv_like.inv f b, (le_map_inv_iff f).2 h, equiv_like.right_inv _ _⟩,
.. order_iso_class.to_order_hom_class } | instance | order_iso_class.to_pseudo_epimorphism_class | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"le_map_inv_iff",
"order_iso_class",
"order_iso_class.to_order_hom_class",
"pseudo_epimorphism_class"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
esakia_hom_class.to_pseudo_epimorphism_class [topological_space α] [preorder α]
[topological_space β] [preorder β] [esakia_hom_class F α β] : pseudo_epimorphism_class F α β | { .. ‹esakia_hom_class F α β› } | instance | esakia_hom_class.to_pseudo_epimorphism_class | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"esakia_hom_class",
"pseudo_epimorphism_class",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_fun_eq_coe {f : pseudo_epimorphism α β} : f.to_fun = (f : α → β) | rfl | lemma | pseudo_epimorphism.to_fun_eq_coe | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"pseudo_epimorphism"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : pseudo_epimorphism α β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | pseudo_epimorphism.ext | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"fun_like.ext",
"pseudo_epimorphism"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : pseudo_epimorphism α β) (f' : α → β) (h : f' = f) :
pseudo_epimorphism α β | ⟨f.to_order_hom.copy f' h, by simpa only [h.symm, to_fun_eq_coe] using f.exists_map_eq_of_map_le'⟩ | def | pseudo_epimorphism.copy | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"pseudo_epimorphism"
] | Copy of a `pseudo_epimorphism` 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 : pseudo_epimorphism α β) (f' : α → β) (h : f' = f) :
⇑(f.copy f' h) = f' | rfl | lemma | pseudo_epimorphism.coe_copy | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"pseudo_epimorphism"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : pseudo_epimorphism α β) (f' : α → β) (h : f' = f) :
f.copy f' h = f | fun_like.ext' h | lemma | pseudo_epimorphism.copy_eq | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"fun_like.ext'",
"pseudo_epimorphism"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : pseudo_epimorphism α α | ⟨order_hom.id, λ a b h, ⟨b, h, rfl⟩⟩ | def | pseudo_epimorphism.id | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"pseudo_epimorphism"
] | `id` as a `pseudo_epimorphism`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(pseudo_epimorphism.id α) = id | rfl | lemma | pseudo_epimorphism.coe_id | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"pseudo_epimorphism.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_id_order_hom : (pseudo_epimorphism.id α : α →o α) = order_hom.id | rfl | lemma | pseudo_epimorphism.coe_id_order_hom | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"order_hom.id",
"pseudo_epimorphism.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (a : α) : pseudo_epimorphism.id α a = a | rfl | lemma | pseudo_epimorphism.id_apply | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"pseudo_epimorphism.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (g : pseudo_epimorphism β γ) (f : pseudo_epimorphism α β) : pseudo_epimorphism α γ | ⟨g.to_order_hom.comp f.to_order_hom, λ a b h₀, begin
obtain ⟨b, h₁, rfl⟩ := g.exists_map_eq_of_map_le' h₀,
obtain ⟨b, h₂, rfl⟩ := f.exists_map_eq_of_map_le' h₁,
exact ⟨b, h₂, rfl⟩,
end⟩ | def | pseudo_epimorphism.comp | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"pseudo_epimorphism"
] | Composition of `pseudo_epimorphism`s as a `pseudo_epimorphism`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (g : pseudo_epimorphism β γ) (f : pseudo_epimorphism α β) :
(g.comp f : α → γ) = g ∘ f | rfl | lemma | pseudo_epimorphism.coe_comp | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"pseudo_epimorphism"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp_order_hom (g : pseudo_epimorphism β γ) (f : pseudo_epimorphism α β) :
(g.comp f : α →o γ) = (g : β →o γ).comp f | rfl | lemma | pseudo_epimorphism.coe_comp_order_hom | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"pseudo_epimorphism"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (g : pseudo_epimorphism β γ) (f : pseudo_epimorphism α β) (a : α) :
(g.comp f) a = g (f a) | rfl | lemma | pseudo_epimorphism.comp_apply | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"pseudo_epimorphism"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (h : pseudo_epimorphism γ δ) (g : pseudo_epimorphism β γ)
(f : pseudo_epimorphism α β) :
(h.comp g).comp f = h.comp (g.comp f) | rfl | lemma | pseudo_epimorphism.comp_assoc | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"pseudo_epimorphism"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : pseudo_epimorphism α β) : f.comp (pseudo_epimorphism.id α) = f | ext $ λ a, rfl | lemma | pseudo_epimorphism.comp_id | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"pseudo_epimorphism",
"pseudo_epimorphism.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : pseudo_epimorphism α β) : (pseudo_epimorphism.id β).comp f = f | ext $ λ a, rfl | lemma | pseudo_epimorphism.id_comp | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"pseudo_epimorphism",
"pseudo_epimorphism.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : pseudo_epimorphism β γ} {f : pseudo_epimorphism 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 | pseudo_epimorphism.cancel_right | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"pseudo_epimorphism"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g : pseudo_epimorphism β γ} {f₁ f₂ : pseudo_epimorphism 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 | pseudo_epimorphism.cancel_left | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"pseudo_epimorphism"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_pseudo_epimorphism (f : esakia_hom α β) : pseudo_epimorphism α β | { ..f } | def | esakia_hom.to_pseudo_epimorphism | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"esakia_hom",
"pseudo_epimorphism"
] | Reinterpret an `esakia_hom` as a `pseudo_epimorphism`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_fun_eq_coe {f : esakia_hom α β} : f.to_fun = (f : α → β) | rfl | lemma | esakia_hom.to_fun_eq_coe | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"esakia_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {f g : esakia_hom α β} (h : ∀ a, f a = g a) : f = g | fun_like.ext f g h | lemma | esakia_hom.ext | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"esakia_hom",
"fun_like.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy (f : esakia_hom α β) (f' : α → β) (h : f' = f) : esakia_hom α β | ⟨f.to_continuous_order_hom.copy f' h,
by simpa only [h.symm, to_fun_eq_coe] using f.exists_map_eq_of_map_le'⟩ | def | esakia_hom.copy | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"esakia_hom"
] | Copy of an `esakia_hom` 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 : esakia_hom α β) (f' : α → β) (h : f' = f) : ⇑(f.copy f' h) = f' | rfl | lemma | esakia_hom.coe_copy | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"esakia_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
copy_eq (f : esakia_hom α β) (f' : α → β) (h : f' = f) : f.copy f' h = f | fun_like.ext' h | lemma | esakia_hom.copy_eq | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"esakia_hom",
"fun_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id : esakia_hom α α | ⟨continuous_order_hom.id α, λ a b h, ⟨b, h, rfl⟩⟩ | def | esakia_hom.id | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"esakia_hom"
] | `id` as an `esakia_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_id : ⇑(esakia_hom.id α) = id | rfl | lemma | esakia_hom.coe_id | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"esakia_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_id_continuous_order_hom :
(esakia_hom.id α : α →Co α) = continuous_order_hom.id α | rfl | lemma | esakia_hom.coe_id_continuous_order_hom | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"continuous_order_hom.id",
"esakia_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_id_pseudo_epimorphism :
(esakia_hom.id α : pseudo_epimorphism α α) = pseudo_epimorphism.id α | rfl | lemma | esakia_hom.coe_id_pseudo_epimorphism | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"esakia_hom.id",
"pseudo_epimorphism",
"pseudo_epimorphism.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_apply (a : α) : esakia_hom.id α a = a | rfl | lemma | esakia_hom.id_apply | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"esakia_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp (g : esakia_hom β γ) (f : esakia_hom α β) : esakia_hom α γ | ⟨g.to_continuous_order_hom.comp f.to_continuous_order_hom, λ a b h₀, begin
obtain ⟨b, h₁, rfl⟩ := g.exists_map_eq_of_map_le' h₀,
obtain ⟨b, h₂, rfl⟩ := f.exists_map_eq_of_map_le' h₁,
exact ⟨b, h₂, rfl⟩,
end⟩ | def | esakia_hom.comp | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"esakia_hom"
] | Composition of `esakia_hom`s as an `esakia_hom`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_comp (g : esakia_hom β γ) (f : esakia_hom α β) : (g.comp f : α → γ) = g ∘ f | rfl | lemma | esakia_hom.coe_comp | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"esakia_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_apply (g : esakia_hom β γ) (f : esakia_hom α β) (a : α) :
(g.comp f) a = g (f a) | rfl | lemma | esakia_hom.comp_apply | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"esakia_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp_continuous_order_hom (g : esakia_hom β γ) (f : esakia_hom α β) :
(g.comp f : α →Co γ) = (g : β →Co γ).comp f | rfl | lemma | esakia_hom.coe_comp_continuous_order_hom | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"esakia_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_comp_pseudo_epimorphism (g : esakia_hom β γ) (f : esakia_hom α β) :
(g.comp f : pseudo_epimorphism α γ) = (g : pseudo_epimorphism β γ).comp f | rfl | lemma | esakia_hom.coe_comp_pseudo_epimorphism | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"esakia_hom",
"pseudo_epimorphism"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_assoc (h : esakia_hom γ δ) (g : esakia_hom β γ) (f : esakia_hom α β) :
(h.comp g).comp f = h.comp (g.comp f) | rfl | lemma | esakia_hom.comp_assoc | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"esakia_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comp_id (f : esakia_hom α β) : f.comp (esakia_hom.id α) = f | ext $ λ a, rfl | lemma | esakia_hom.comp_id | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"esakia_hom",
"esakia_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
id_comp (f : esakia_hom α β) : (esakia_hom.id β).comp f = f | ext $ λ a, rfl | lemma | esakia_hom.id_comp | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"esakia_hom",
"esakia_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_right {g₁ g₂ : esakia_hom β γ} {f : esakia_hom 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 | esakia_hom.cancel_right | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"esakia_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
cancel_left {g : esakia_hom β γ} {f₁ f₂ : esakia_hom 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 | esakia_hom.cancel_left | topology.order.hom | src/topology/order/hom/esakia.lean | [
"order.hom.bounded",
"topology.order.hom.basic"
] | [
"esakia_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closeds (α : Type*) [topological_space α] | (carrier : set α)
(closed' : is_closed carrier) | structure | topological_space.closeds | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"is_closed",
"topological_space"
] | The type of closed subsets of a topological space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closed (s : closeds α) : is_closed (s : set α) | s.closed' | lemma | topological_space.closeds.closed | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {s t : closeds α} (h : (s : set α) = t) : s = t | set_like.ext' h | lemma | topological_space.closeds.ext | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"set_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk (s : set α) (h) : (mk s h : set α) = s | rfl | lemma | topological_space.closeds.coe_mk | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure (s : set α) : closeds α | ⟨closure s, is_closed_closure⟩ | def | topological_space.closeds.closure | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"closure"
] | The closure of a set, as an element of `closeds`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gc : galois_connection closeds.closure (coe : closeds α → set α) | λ s U, ⟨subset_closure.trans, λ h, closure_minimal h U.closed⟩ | lemma | topological_space.closeds.gc | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"closure_minimal",
"galois_connection"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gi : galois_insertion (@closeds.closure α _) coe | { choice := λ s hs, ⟨s, closure_eq_iff_is_closed.1 $ hs.antisymm subset_closure⟩,
gc := gc,
le_l_u := λ _, subset_closure,
choice_eq := λ s hs, set_like.coe_injective $ subset_closure.antisymm hs } | def | topological_space.closeds.gi | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"galois_insertion",
"set_like.coe_injective",
"subset_closure"
] | The galois coinsertion between sets and opens. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_sup (s t : closeds α) : (↑(s ⊔ t) : set α) = s ∪ t | rfl | lemma | topological_space.closeds.coe_sup | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inf (s t : closeds α) : (↑(s ⊓ t) : set α) = s ∩ t | rfl | lemma | topological_space.closeds.coe_inf | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_top : (↑(⊤ : closeds α) : set α) = univ | rfl | lemma | topological_space.closeds.coe_top | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bot : (↑(⊥ : closeds α) : set α) = ∅ | rfl | lemma | topological_space.closeds.coe_bot | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_Inf {S : set (closeds α)} : (↑(Inf S) : set α) = ⋂ i ∈ S, ↑i | rfl | lemma | topological_space.closeds.coe_Inf | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_finset_sup (f : ι → closeds α) (s : finset ι) :
(↑(s.sup f) : set α) = s.sup (coe ∘ f) | map_finset_sup (⟨⟨coe, coe_sup⟩, coe_bot⟩ : sup_bot_hom (closeds α) (set α)) _ _ | lemma | topological_space.closeds.coe_finset_sup | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"finset",
"map_finset_sup",
"sup_bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_finset_inf (f : ι → closeds α) (s : finset ι) :
(↑(s.inf f) : set α) = s.inf (coe ∘ f) | map_finset_inf (⟨⟨coe, coe_inf⟩, coe_top⟩ : inf_top_hom (closeds α) (set α)) _ _ | lemma | topological_space.closeds.coe_finset_inf | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"finset",
"inf_top_hom",
"map_finset_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_def {ι} (s : ι → closeds α) : (⨅ i, s i) = ⟨⋂ i, s i, is_closed_Inter $ λ i, (s i).2⟩ | by { ext, simp only [infi, coe_Inf, bInter_range], refl } | lemma | topological_space.closeds.infi_def | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"infi",
"is_closed_Inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
infi_mk {ι} (s : ι → set α) (h : ∀ i, is_closed (s i)) :
(⨅ i, ⟨s i, h i⟩ : closeds α) = ⟨⋂ i, s i, is_closed_Inter h⟩ | by simp [infi_def] | lemma | topological_space.closeds.infi_mk | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"is_closed",
"is_closed_Inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_infi {ι} (s : ι → closeds α) :
((⨅ i, s i : closeds α) : set α) = ⋂ i, s i | by simp [infi_def] | lemma | topological_space.closeds.coe_infi | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_infi {ι} {x : α} {s : ι → closeds α} : x ∈ infi s ↔ ∀ i, x ∈ s i | by simp [←set_like.mem_coe] | lemma | topological_space.closeds.mem_infi | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_Inf {S : set (closeds α)} {x : α} : x ∈ Inf S ↔ ∀ s ∈ S, x ∈ s | by simp_rw [Inf_eq_infi, mem_infi] | lemma | topological_space.closeds.mem_Inf | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"Inf_eq_infi"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
singleton [t1_space α] (x : α) : closeds α | ⟨{x}, is_closed_singleton⟩ | def | topological_space.closeds.singleton | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"t1_space"
] | The term of `closeds α` corresponding to a singleton. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closeds.compl (s : closeds α) : opens α | ⟨sᶜ, s.2.is_open_compl⟩ | def | topological_space.closeds.compl | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [] | The complement of a closed set as an open set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
opens.compl (s : opens α) : closeds α | ⟨sᶜ, s.2.is_closed_compl⟩ | def | topological_space.opens.compl | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [] | The complement of an open set as a closed set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closeds.compl_compl (s : closeds α) : s.compl.compl = s | closeds.ext (compl_compl s) | lemma | topological_space.closeds.compl_compl | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"compl_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opens.compl_compl (s : opens α) : s.compl.compl = s | opens.ext (compl_compl s) | lemma | topological_space.opens.compl_compl | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"compl_compl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closeds.compl_bijective : function.bijective (@closeds.compl α _) | function.bijective_iff_has_inverse.mpr ⟨opens.compl, closeds.compl_compl, opens.compl_compl⟩ | lemma | topological_space.closeds.compl_bijective | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opens.compl_bijective : function.bijective (@opens.compl α _) | function.bijective_iff_has_inverse.mpr ⟨closeds.compl, opens.compl_compl, closeds.compl_compl⟩ | lemma | topological_space.opens.compl_bijective | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closeds.compl_order_iso : closeds α ≃o (opens α)ᵒᵈ | { to_fun := order_dual.to_dual ∘ closeds.compl,
inv_fun := opens.compl ∘ order_dual.of_dual,
left_inv := λ s, by simp [closeds.compl_compl],
right_inv := λ s, by simp [opens.compl_compl],
map_rel_iff' := λ s t, by simpa only [equiv.coe_fn_mk, function.comp_app,
order_dual.to_dual_le_to_dual] using compl_sub... | def | topological_space.closeds.compl_order_iso | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"equiv.coe_fn_mk",
"inv_fun",
"order_dual.of_dual",
"order_dual.to_dual",
"order_dual.to_dual_le_to_dual"
] | `closeds.compl` as an `order_iso` to the order dual of `opens α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
opens.compl_order_iso : opens α ≃o (closeds α)ᵒᵈ | { to_fun := order_dual.to_dual ∘ opens.compl,
inv_fun := closeds.compl ∘ order_dual.of_dual,
left_inv := λ s, by simp [opens.compl_compl],
right_inv := λ s, by simp [closeds.compl_compl],
map_rel_iff' := λ s t, by simpa only [equiv.coe_fn_mk, function.comp_app,
order_dual.to_dual_le_to_dual] using compl_sub... | def | topological_space.opens.compl_order_iso | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"equiv.coe_fn_mk",
"inv_fun",
"order_dual.of_dual",
"order_dual.to_dual",
"order_dual.to_dual_le_to_dual"
] | `opens.compl` as an `order_iso` to the order dual of `closeds α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
closeds.is_atom_iff [t1_space α] {s : closeds α} : is_atom s ↔ ∃ x, s = closeds.singleton x | begin
have : is_atom (s : set α) ↔ is_atom s,
{ refine closeds.gi.is_atom_iff' rfl (λ t ht, _) s,
obtain ⟨x, rfl⟩ := t.is_atom_iff.mp ht,
exact closure_singleton },
simpa only [← this, (s : set α).is_atom_iff, set_like.ext_iff, set.ext_iff]
end | lemma | topological_space.closeds.is_atom_iff | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"closure_singleton",
"is_atom",
"is_atom_iff",
"set.ext_iff",
"set_like.ext_iff",
"t1_space"
] | in a `t1_space`, atoms of `closeds α` are precisely the `closeds.singleton`s. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
opens.is_coatom_iff [t1_space α] {s : opens α} :
is_coatom s ↔ ∃ x, s = (closeds.singleton x).compl | begin
rw [←s.compl_compl, ←is_atom_dual_iff_is_coatom],
change is_atom (closeds.compl_order_iso α s.compl) ↔ _,
rw [(closeds.compl_order_iso α).is_atom_iff, closeds.is_atom_iff],
congrm ∃ x, _,
exact closeds.compl_bijective.injective.eq_iff.symm,
end | lemma | topological_space.opens.is_coatom_iff | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"is_atom",
"is_atom_iff",
"is_coatom",
"t1_space"
] | in a `t1_space`, coatoms of `opens α` are precisely complements of singletons:
`(closeds.singleton x).compl`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
clopens (α : Type*) [topological_space α] | (carrier : set α)
(clopen' : is_clopen carrier) | structure | topological_space.clopens | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"is_clopen",
"topological_space"
] | The type of clopen sets of a topological space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
clopen (s : clopens α) : is_clopen (s : set α) | s.clopen' | lemma | topological_space.clopens.clopen | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"is_clopen"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_opens (s : clopens α) : opens α | ⟨s, s.clopen.is_open⟩ | def | topological_space.clopens.to_opens | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [] | Reinterpret a compact open as an open. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {s t : clopens α} (h : (s : set α) = t) : s = t | set_like.ext' h | lemma | topological_space.clopens.ext | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [
"set_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sup (s t : clopens α) : (↑(s ⊔ t) : set α) = s ∪ t | rfl | lemma | topological_space.clopens.coe_sup | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inf (s t : clopens α) : (↑(s ⊓ t) : set α) = s ∩ t | rfl | lemma | topological_space.clopens.coe_inf | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_top : (↑(⊤ : clopens α) : set α) = univ | rfl | lemma | topological_space.clopens.coe_top | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bot : (↑(⊥ : clopens α) : set α) = ∅ | rfl | lemma | topological_space.clopens.coe_bot | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sdiff (s t : clopens α) : (↑(s \ t) : set α) = s \ t | rfl | lemma | topological_space.clopens.coe_sdiff | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_compl (s : clopens α) : (↑sᶜ : set α) = sᶜ | rfl | lemma | topological_space.clopens.coe_compl | topology.sets | src/topology/sets/closeds.lean | [
"topology.sets.opens"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compacts (α : Type*) [topological_space α] | (carrier : set α)
(is_compact' : is_compact carrier) | structure | topological_space.compacts | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"is_compact",
"topological_space"
] | The type of compact sets of a topological space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact (s : compacts α) : is_compact (s : set α) | s.is_compact' | lemma | topological_space.compacts.is_compact | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {s t : compacts α} (h : (s : set α) = t) : s = t | set_like.ext' h | lemma | topological_space.compacts.ext | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"set_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
carrier_eq_coe (s : compacts α) : s.carrier = s | rfl | lemma | topological_space.compacts.carrier_eq_coe | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sup (s t : compacts α) : (↑(s ⊔ t) : set α) = s ∪ t | rfl | lemma | topological_space.compacts.coe_sup | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inf [t2_space α] (s t : compacts α) : (↑(s ⊓ t) : set α) = s ∩ t | rfl | lemma | topological_space.compacts.coe_inf | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_top [compact_space α] : (↑(⊤ : compacts α) : set α) = univ | rfl | lemma | topological_space.compacts.coe_top | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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