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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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embedding.second_countable_topology [second_countable_topology β]
(hf : embedding f) : second_countable_topology α | hf.1.second_countable_topology | lemma | embedding.second_countable_topology | topology | src/topology/bases.lean | [
"topology.constructions",
"topology.continuous_on"
] | [
"embedding"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_space (α : Type u) | (is_open : set α → Prop)
(is_open_univ : is_open univ)
(is_open_inter : ∀s t, is_open s → is_open t → is_open (s ∩ t))
(is_open_sUnion : ∀s, (∀t∈s, is_open t) → is_open (⋃₀ s)) | class | topological_space | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"is_open_sUnion",
"is_open_univ"
] | A topology on `α`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
topological_space.of_closed {α : Type u} (T : set (set α))
(empty_mem : ∅ ∈ T) (sInter_mem : ∀ A ⊆ T, ⋂₀ A ∈ T) (union_mem : ∀ A B ∈ T, A ∪ B ∈ T) :
topological_space α | { is_open := λ X, Xᶜ ∈ T,
is_open_univ := by simp [empty_mem],
is_open_inter := λ s t hs ht, by simpa only [compl_inter] using union_mem sᶜ hs tᶜ ht,
is_open_sUnion := λ s hs,
by rw set.compl_sUnion; exact sInter_mem (compl '' s)
(λ z ⟨y, hy, hz⟩, by simpa [hz.symm] using hs y hy) } | def | topological_space.of_closed | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"is_open_sUnion",
"is_open_univ",
"set.compl_sUnion",
"topological_space"
] | A constructor for topologies by specifying the closed sets,
and showing that they satisfy the appropriate conditions. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open [topological_space α] (s : set α) : Prop | @topological_space.is_open _ ‹_› s | def | is_open | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"topological_space"
] | `is_open s` means that `s` is open in the ambient topological space on `α` | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_mk {p h₁ h₂ h₃} {s : set α} : is_open[⟨p, h₁, h₂, h₃⟩] s ↔ p s | iff.rfl | lemma | is_open_mk | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_space_eq {f g : topological_space α} (h : is_open[f] = is_open[g]) : f = g | by unfreezingI { cases f, cases g, congr, exact h } | lemma | topological_space_eq | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_univ : is_open (univ : set α) | topological_space.is_open_univ | lemma | is_open_univ | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.inter (h₁ : is_open s₁) (h₂ : is_open s₂) : is_open (s₁ ∩ s₂) | topological_space.is_open_inter s₁ s₂ h₁ h₂ | lemma | is_open.inter | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_sUnion {s : set (set α)} (h : ∀t ∈ s, is_open t) : is_open (⋃₀ s) | topological_space.is_open_sUnion s h | lemma | is_open_sUnion | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
topological_space_eq_iff {t t' : topological_space α} :
t = t' ↔ ∀ s, is_open[t] s ↔ is_open[t'] s | ⟨λ h s, h ▸ iff.rfl, λ h, by { ext, exact h _ }⟩ | lemma | topological_space_eq_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_fold {s : set α} {t : topological_space α} : t.is_open s = is_open[t] s | rfl | lemma | is_open_fold | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_Union {f : ι → set α} (h : ∀i, is_open (f i)) : is_open (⋃i, f i) | is_open_sUnion $ by rintro _ ⟨i, rfl⟩; exact h i | lemma | is_open_Union | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"is_open_sUnion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_bUnion {s : set β} {f : β → set α} (h : ∀i∈s, is_open (f i)) :
is_open (⋃i∈s, f i) | is_open_Union $ assume i, is_open_Union $ assume hi, h i hi | lemma | is_open_bUnion | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"is_open_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.union (h₁ : is_open s₁) (h₂ : is_open s₂) : is_open (s₁ ∪ s₂) | by rw union_eq_Union; exact is_open_Union (bool.forall_bool.2 ⟨h₂, h₁⟩) | lemma | is_open.union | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"is_open_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_empty : is_open (∅ : set α) | by rw ← sUnion_empty; exact is_open_sUnion (assume a, false.elim) | lemma | is_open_empty | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"is_open_sUnion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_sInter {s : set (set α)} (hs : s.finite) : (∀t ∈ s, is_open t) → is_open (⋂₀ s) | finite.induction_on hs (λ _, by rw sInter_empty; exact is_open_univ) $
λ a s has hs ih h, by rw sInter_insert; exact
is_open.inter (h _ $ mem_insert _ _) (ih $ λ t, h t ∘ mem_insert_of_mem _) | lemma | is_open_sInter | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"ih",
"is_open",
"is_open.inter",
"is_open_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_bInter {s : set β} {f : β → set α} (hs : s.finite) :
(∀i∈s, is_open (f i)) → is_open (⋂i∈s, f i) | finite.induction_on hs
(λ _, by rw bInter_empty; exact is_open_univ)
(λ a s has hs ih h, by rw bInter_insert; exact
is_open.inter (h a (mem_insert _ _)) (ih (λ i hi, h i (mem_insert_of_mem _ hi)))) | lemma | is_open_bInter | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"ih",
"is_open",
"is_open.inter",
"is_open_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_Inter [finite ι] {s : ι → set α} (h : ∀ i, is_open (s i)) : is_open (⋂ i, s i) | is_open_sInter (finite_range _) (forall_range_iff.2 h) | lemma | is_open_Inter | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"finite",
"is_open",
"is_open_sInter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_bInter_finset {s : finset β} {f : β → set α} (h : ∀ i ∈ s, is_open (f i)) :
is_open (⋂ i ∈ s, f i) | is_open_bInter (to_finite _) h | lemma | is_open_bInter_finset | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"finset",
"is_open",
"is_open_bInter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_const {p : Prop} : is_open {a : α | p} | by_cases
(assume : p, begin simp only [this]; exact is_open_univ end)
(assume : ¬ p, begin simp only [this]; exact is_open_empty end) | lemma | is_open_const | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"is_open_empty",
"is_open_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.and : is_open {a | p₁ a} → is_open {a | p₂ a} → is_open {a | p₁ a ∧ p₂ a} | is_open.inter | lemma | is_open.and | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"is_open.inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed (s : set α) : Prop | (is_open_compl : is_open sᶜ) | class | is_closed | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open"
] | A set is closed if its complement is open | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_open_compl_iff {s : set α} : is_open sᶜ ↔ is_closed s | ⟨λ h, ⟨h⟩, λ h, h.is_open_compl⟩ | lemma | is_open_compl_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_closed",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_empty : is_closed (∅ : set α) | by { rw [← is_open_compl_iff, compl_empty], exact is_open_univ } | lemma | is_closed_empty | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_closed",
"is_open_compl_iff",
"is_open_univ"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_univ : is_closed (univ : set α) | by { rw [← is_open_compl_iff, compl_univ], exact is_open_empty } | lemma | is_closed_univ | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_closed",
"is_open_compl_iff",
"is_open_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.union : is_closed s₁ → is_closed s₂ → is_closed (s₁ ∪ s₂) | λ h₁ h₂, by { rw [← is_open_compl_iff] at *, rw compl_union, exact is_open.inter h₁ h₂ } | lemma | is_closed.union | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_closed",
"is_open.inter",
"is_open_compl_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_sInter {s : set (set α)} : (∀t ∈ s, is_closed t) → is_closed (⋂₀ s) | by simpa only [← is_open_compl_iff, compl_sInter, sUnion_image] using is_open_bUnion | lemma | is_closed_sInter | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_closed",
"is_open_bUnion",
"is_open_compl_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_Inter {f : ι → set α} (h : ∀i, is_closed (f i)) : is_closed (⋂i, f i ) | is_closed_sInter $ assume t ⟨i, (heq : f i = t)⟩, heq ▸ h i | lemma | is_closed_Inter | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_closed",
"is_closed_sInter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_bInter {s : set β} {f : β → set α} (h : ∀ i ∈ s, is_closed (f i)) :
is_closed (⋂ i ∈ s, f i) | is_closed_Inter $ λ i, is_closed_Inter $ h i | lemma | is_closed_bInter | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_closed",
"is_closed_Inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_compl_iff {s : set α} : is_closed sᶜ ↔ is_open s | by rw [←is_open_compl_iff, compl_compl] | lemma | is_closed_compl_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"compl_compl",
"is_closed",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.is_closed_compl {s : set α} (hs : is_open s) : is_closed sᶜ | is_closed_compl_iff.2 hs | lemma | is_open.is_closed_compl | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_closed",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.sdiff {s t : set α} (h₁ : is_open s) (h₂ : is_closed t) : is_open (s \ t) | is_open.inter h₁ $ is_open_compl_iff.mpr h₂ | lemma | is_open.sdiff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_closed",
"is_open",
"is_open.inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.inter (h₁ : is_closed s₁) (h₂ : is_closed s₂) : is_closed (s₁ ∩ s₂) | by { rw [← is_open_compl_iff] at *, rw compl_inter, exact is_open.union h₁ h₂ } | lemma | is_closed.inter | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_closed",
"is_open.union",
"is_open_compl_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.sdiff {s t : set α} (h₁ : is_closed s) (h₂ : is_open t) : is_closed (s \ t) | is_closed.inter h₁ (is_closed_compl_iff.mpr h₂) | lemma | is_closed.sdiff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_closed",
"is_closed.inter",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_bUnion {s : set β} {f : β → set α} (hs : s.finite) :
(∀i∈s, is_closed (f i)) → is_closed (⋃i∈s, f i) | finite.induction_on hs
(λ _, by rw bUnion_empty; exact is_closed_empty)
(λ a s has hs ih h, by rw bUnion_insert; exact
is_closed.union (h a (mem_insert _ _)) (ih (λ i hi, h i (mem_insert_of_mem _ hi)))) | lemma | is_closed_bUnion | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"ih",
"is_closed",
"is_closed.union",
"is_closed_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_Union [finite ι] {s : ι → set α} (h : ∀ i, is_closed (s i)) :
is_closed (⋃ i, s i) | by { simp only [← is_open_compl_iff, compl_Union] at *, exact is_open_Inter h } | lemma | is_closed_Union | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"finite",
"is_closed",
"is_open_Inter",
"is_open_compl_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_imp {p q : α → Prop} (hp : is_open {x | p x})
(hq : is_closed {x | q x}) : is_closed {x | p x → q x} | have {x | p x → q x} = {x | p x}ᶜ ∪ {x | q x}, from set.ext $ λ x, imp_iff_not_or,
by rw [this]; exact is_closed.union (is_closed_compl_iff.mpr hp) hq | lemma | is_closed_imp | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"imp_iff_not_or",
"is_closed",
"is_closed.union",
"is_open",
"set.ext"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.not : is_closed {a | p a} → is_open {a | ¬ p a} | is_open_compl_iff.mpr | lemma | is_closed.not | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_closed",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior (s : set α) : set α | ⋃₀ {t | is_open t ∧ t ⊆ s} | def | interior | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open"
] | The interior of a set `s` is the largest open subset of `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_interior {s : set α} {x : α} :
x ∈ interior s ↔ ∃ t ⊆ s, is_open t ∧ x ∈ t | by simp only [interior, mem_sUnion, mem_set_of_eq, exists_prop, and_assoc, and.left_comm] | lemma | mem_interior | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"exists_prop",
"interior",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_interior {s : set α} : is_open (interior s) | is_open_sUnion $ assume t ⟨h₁, h₂⟩, h₁ | lemma | is_open_interior | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"is_open",
"is_open_sUnion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_subset {s : set α} : interior s ⊆ s | sUnion_subset $ assume t ⟨h₁, h₂⟩, h₂ | lemma | interior_subset | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_maximal {s t : set α} (h₁ : t ⊆ s) (h₂ : is_open t) : t ⊆ interior s | subset_sUnion_of_mem ⟨h₂, h₁⟩ | lemma | interior_maximal | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.interior_eq {s : set α} (h : is_open s) : interior s = s | subset.antisymm interior_subset (interior_maximal (subset.refl s) h) | lemma | is_open.interior_eq | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"interior_maximal",
"interior_subset",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_eq_iff_is_open {s : set α} : interior s = s ↔ is_open s | ⟨assume h, h ▸ is_open_interior, is_open.interior_eq⟩ | lemma | interior_eq_iff_is_open | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"is_open",
"is_open_interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_interior_iff_is_open {s : set α} : s ⊆ interior s ↔ is_open s | by simp only [interior_eq_iff_is_open.symm, subset.antisymm_iff, interior_subset, true_and] | lemma | subset_interior_iff_is_open | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"interior_subset",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.subset_interior_iff {s t : set α} (h₁ : is_open s) :
s ⊆ interior t ↔ s ⊆ t | ⟨assume h, subset.trans h interior_subset, assume h₂, interior_maximal h₂ h₁⟩ | lemma | is_open.subset_interior_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"interior_maximal",
"interior_subset",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_interior_iff {s t : set α} : t ⊆ interior s ↔ ∃ U, is_open U ∧ t ⊆ U ∧ U ⊆ s | ⟨λ h, ⟨interior s, is_open_interior, h, interior_subset⟩,
λ ⟨U, hU, htU, hUs⟩, htU.trans (interior_maximal hUs hU)⟩ | lemma | subset_interior_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"interior_maximal",
"is_open",
"is_open_interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_mono {s t : set α} (h : s ⊆ t) : interior s ⊆ interior t | interior_maximal (subset.trans interior_subset h) is_open_interior | lemma | interior_mono | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"interior_maximal",
"interior_subset",
"is_open_interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_empty : interior (∅ : set α) = ∅ | is_open_empty.interior_eq | lemma | interior_empty | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_univ : interior (univ : set α) = univ | is_open_univ.interior_eq | lemma | interior_univ | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_eq_univ {s : set α} : interior s = univ ↔ s = univ | ⟨λ h, univ_subset_iff.mp $ h.symm.trans_le interior_subset, λ h, h.symm ▸ interior_univ⟩ | lemma | interior_eq_univ | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"interior_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_interior {s : set α} : interior (interior s) = interior s | is_open_interior.interior_eq | lemma | interior_interior | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_inter {s t : set α} : interior (s ∩ t) = interior s ∩ interior t | subset.antisymm
(subset_inter (interior_mono $ inter_subset_left s t) (interior_mono $ inter_subset_right s t))
(interior_maximal (inter_subset_inter interior_subset interior_subset) $
is_open.inter is_open_interior is_open_interior) | lemma | interior_inter | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"interior_maximal",
"interior_mono",
"interior_subset",
"is_open.inter",
"is_open_interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.interior_Inter {ι : Type*} (s : finset ι) (f : ι → set α) :
interior (⋂ i ∈ s, f i) = ⋂ i ∈ s, interior (f i) | begin
classical,
refine s.induction_on (by simp) _,
intros i s h₁ h₂,
simp [h₂],
end | lemma | finset.interior_Inter | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"finset",
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_Inter {ι : Type*} [finite ι] (f : ι → set α) :
interior (⋂ i, f i) = ⋂ i, interior (f i) | by { casesI nonempty_fintype ι, convert finset.univ.interior_Inter f; simp } | lemma | interior_Inter | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"finite",
"interior",
"nonempty_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_union_is_closed_of_interior_empty {s t : set α} (h₁ : is_closed s)
(h₂ : interior t = ∅) :
interior (s ∪ t) = interior s | have interior (s ∪ t) ⊆ s, from
assume x ⟨u, ⟨(hu₁ : is_open u), (hu₂ : u ⊆ s ∪ t)⟩, (hx₁ : x ∈ u)⟩,
classical.by_contradiction $ assume hx₂ : x ∉ s,
have u \ s ⊆ t,
from assume x ⟨h₁, h₂⟩, or.resolve_left (hu₂ h₁) h₂,
have u \ s ⊆ interior t,
by rwa (is_open.sdiff hu₁ h₁).subset_interior_iff,
... | lemma | interior_union_is_closed_of_interior_empty | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"interior_maximal",
"interior_mono",
"is_closed",
"is_open",
"is_open.sdiff",
"is_open_interior",
"subset_interior_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_iff_forall_mem_open : is_open s ↔ ∀ x ∈ s, ∃ t ⊆ s, is_open t ∧ x ∈ t | by rw ← subset_interior_iff_is_open; simp only [subset_def, mem_interior] | lemma | is_open_iff_forall_mem_open | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_open",
"mem_interior",
"subset_interior_iff_is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_Inter_subset (s : ι → set α) : interior (⋂ i, s i) ⊆ ⋂ i, interior (s i) | subset_Inter $ λ i, interior_mono $ Inter_subset _ _ | lemma | interior_Inter_subset | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"interior_mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_Inter₂_subset (p : ι → Sort*) (s : Π i, p i → set α) :
interior (⋂ i j, s i j) ⊆ ⋂ i j, interior (s i j) | (interior_Inter_subset _).trans $ Inter_mono $ λ i, interior_Inter_subset _ | lemma | interior_Inter₂_subset | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"interior_Inter_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_sInter_subset (S : set (set α)) : interior (⋂₀ S) ⊆ ⋂ s ∈ S, interior s | calc interior (⋂₀ S) = interior (⋂ s ∈ S, s) : by rw sInter_eq_bInter
... ⊆ ⋂ s ∈ S, interior s : interior_Inter₂_subset _ _ | lemma | interior_sInter_subset | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"interior",
"interior_Inter₂_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure (s : set α) : set α | ⋂₀ {t | is_closed t ∧ s ⊆ t} | def | closure | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"is_closed"
] | The closure of `s` is the smallest closed set containing `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_closed_closure {s : set α} : is_closed (closure s) | is_closed_sInter $ assume t ⟨h₁, h₂⟩, h₁ | lemma | is_closed_closure | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"is_closed",
"is_closed_sInter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subset_closure {s : set α} : s ⊆ closure s | subset_sInter $ assume t ⟨h₁, h₂⟩, h₂ | lemma | subset_closure | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_mem_of_not_mem_closure {s : set α} {P : α} (hP : P ∉ closure s) : P ∉ s | λ h, hP (subset_closure h) | lemma | not_mem_of_not_mem_closure | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_minimal {s t : set α} (h₁ : s ⊆ t) (h₂ : is_closed t) : closure s ⊆ t | sInter_subset_of_mem ⟨h₂, h₁⟩ | lemma | closure_minimal | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint.closure_left {s t : set α} (hd : disjoint s t) (ht : is_open t) :
disjoint (closure s) t | disjoint_compl_left.mono_left $ closure_minimal hd.subset_compl_right ht.is_closed_compl | lemma | disjoint.closure_left | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"closure_minimal",
"disjoint",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
disjoint.closure_right {s t : set α} (hd : disjoint s t) (hs : is_open s) :
disjoint s (closure t) | (hd.symm.closure_left hs).symm | lemma | disjoint.closure_right | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"disjoint",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.closure_eq {s : set α} (h : is_closed s) : closure s = s | subset.antisymm (closure_minimal (subset.refl s) h) subset_closure | lemma | is_closed.closure_eq | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"closure_minimal",
"is_closed",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.closure_subset {s : set α} (hs : is_closed s) : closure s ⊆ s | closure_minimal (subset.refl _) hs | lemma | is_closed.closure_subset | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"closure_minimal",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.closure_subset_iff {s t : set α} (h₁ : is_closed t) :
closure s ⊆ t ↔ s ⊆ t | ⟨subset.trans subset_closure, assume h, closure_minimal h h₁⟩ | lemma | is_closed.closure_subset_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"closure_minimal",
"is_closed",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed.mem_iff_closure_subset {s : set α} (hs : is_closed s) {x : α} :
x ∈ s ↔ closure ({x} : set α) ⊆ s | (hs.closure_subset_iff.trans set.singleton_subset_iff).symm | lemma | is_closed.mem_iff_closure_subset | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"is_closed",
"set.singleton_subset_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_mono {s t : set α} (h : s ⊆ t) : closure s ⊆ closure t | closure_minimal (subset.trans h subset_closure) is_closed_closure | lemma | closure_mono | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"closure_minimal",
"is_closed_closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
monotone_closure (α : Type*) [topological_space α] : monotone (@closure α _) | λ _ _, closure_mono | lemma | monotone_closure | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"closure_mono",
"monotone",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
diff_subset_closure_iff {s t : set α} :
s \ t ⊆ closure t ↔ s ⊆ closure t | by rw [diff_subset_iff, union_eq_self_of_subset_left subset_closure] | lemma | diff_subset_closure_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_inter_subset_inter_closure (s t : set α) :
closure (s ∩ t) ⊆ closure s ∩ closure t | (monotone_closure α).map_inf_le s t | lemma | closure_inter_subset_inter_closure | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"monotone_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_of_closure_subset {s : set α} (h : closure s ⊆ s) : is_closed s | by rw subset.antisymm subset_closure h; exact is_closed_closure | lemma | is_closed_of_closure_subset | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"is_closed",
"is_closed_closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_eq_iff_is_closed {s : set α} : closure s = s ↔ is_closed s | ⟨assume h, h ▸ is_closed_closure, is_closed.closure_eq⟩ | lemma | closure_eq_iff_is_closed | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"is_closed",
"is_closed_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_subset_iff_is_closed {s : set α} : closure s ⊆ s ↔ is_closed s | ⟨is_closed_of_closure_subset, is_closed.closure_subset⟩ | lemma | closure_subset_iff_is_closed | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_empty : closure (∅ : set α) = ∅ | is_closed_empty.closure_eq | lemma | closure_empty | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_empty_iff (s : set α) : closure s = ∅ ↔ s = ∅ | ⟨subset_eq_empty subset_closure, λ h, h.symm ▸ closure_empty⟩ | lemma | closure_empty_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_nonempty_iff {s : set α} : (closure s).nonempty ↔ s.nonempty | by simp only [nonempty_iff_ne_empty, ne.def, closure_empty_iff] | lemma | closure_nonempty_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"closure_empty_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_univ : closure (univ : set α) = univ | is_closed_univ.closure_eq | lemma | closure_univ | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_closure {s : set α} : closure (closure s) = closure s | is_closed_closure.closure_eq | lemma | closure_closure | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_union {s t : set α} : closure (s ∪ t) = closure s ∪ closure t | subset.antisymm
(closure_minimal (union_subset_union subset_closure subset_closure) $
is_closed.union is_closed_closure is_closed_closure)
((monotone_closure α).le_map_sup s t) | lemma | closure_union | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"closure_minimal",
"is_closed.union",
"is_closed_closure",
"monotone_closure",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
finset.closure_bUnion {ι : Type*} (s : finset ι) (f : ι → set α) :
closure (⋃ i ∈ s, f i) = ⋃ i ∈ s, closure (f i) | begin
classical,
refine s.induction_on (by simp) _,
intros i s h₁ h₂,
simp [h₂],
end | lemma | finset.closure_bUnion | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"finset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_Union {ι : Type*} [finite ι] (f : ι → set α) :
closure (⋃ i, f i) = ⋃ i, closure (f i) | by { casesI nonempty_fintype ι, convert finset.univ.closure_bUnion f; simp } | lemma | closure_Union | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"finite",
"nonempty_fintype"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_subset_closure {s : set α} : interior s ⊆ closure s | subset.trans interior_subset subset_closure | lemma | interior_subset_closure | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"interior",
"interior_subset",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_eq_compl_interior_compl {s : set α} : closure s = (interior sᶜ)ᶜ | begin
rw [interior, closure, compl_sUnion, compl_image_set_of],
simp only [compl_subset_compl, is_open_compl_iff],
end | lemma | closure_eq_compl_interior_compl | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"interior",
"is_open_compl_iff"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
interior_compl {s : set α} : interior sᶜ = (closure s)ᶜ | by simp [closure_eq_compl_interior_compl] | lemma | interior_compl | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"closure_eq_compl_interior_compl",
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_compl {s : set α} : closure sᶜ = (interior s)ᶜ | by simp [closure_eq_compl_interior_compl] | lemma | closure_compl | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"closure_eq_compl_interior_compl",
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_closure_iff {s : set α} {a : α} :
a ∈ closure s ↔ ∀ o, is_open o → a ∈ o → (o ∩ s).nonempty | ⟨λ h o oo ao, classical.by_contradiction $ λ os,
have s ⊆ oᶜ, from λ x xs xo, os ⟨x, xo, xs⟩,
closure_minimal this (is_closed_compl_iff.2 oo) h ao,
λ H c ⟨h₁, h₂⟩, classical.by_contradiction $ λ nc,
let ⟨x, hc, hs⟩ := (H _ h₁.is_open_compl nc) in hc (h₂ hs)⟩ | theorem | mem_closure_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"closure_minimal",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
closure_inter_open_nonempty_iff {s t : set α} (h : is_open t) :
(closure s ∩ t).nonempty ↔ (s ∩ t).nonempty | ⟨λ ⟨x, hxcs, hxt⟩, inter_comm t s ▸ mem_closure_iff.1 hxcs t h hxt,
λ h, h.mono $ inf_le_inf_right t subset_closure⟩ | lemma | closure_inter_open_nonempty_iff | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"inf_le_inf_right",
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.le_lift'_closure (l : filter α) : l ≤ l.lift' closure | le_lift'.2 $ λ s hs, mem_of_superset hs subset_closure | lemma | filter.le_lift'_closure | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"filter",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.lift'_closure {l : filter α} {p : ι → Prop} {s : ι → set α}
(h : l.has_basis p s) :
(l.lift' closure).has_basis p (λ i, closure (s i)) | h.lift' (monotone_closure α) | lemma | filter.has_basis.lift'_closure | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"filter",
"monotone_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.has_basis.lift'_closure_eq_self {l : filter α} {p : ι → Prop} {s : ι → set α}
(h : l.has_basis p s) (hc : ∀ i, p i → is_closed (s i)) :
l.lift' closure = l | le_antisymm (h.ge_iff.2 $ λ i hi, (hc i hi).closure_eq ▸ mem_lift' (h.mem_of_mem hi))
l.le_lift'_closure | lemma | filter.has_basis.lift'_closure_eq_self | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"filter",
"is_closed"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
filter.lift'_closure_eq_bot {l : filter α} : l.lift' closure = ⊥ ↔ l = ⊥ | ⟨λ h, bot_unique $ h ▸ l.le_lift'_closure,
λ h, h.symm ▸ by rw [lift'_bot (monotone_closure _), closure_empty, principal_empty]⟩ | lemma | filter.lift'_closure_eq_bot | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"bot_unique",
"closure",
"closure_empty",
"filter",
"monotone_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense (s : set α) : Prop | ∀ x, x ∈ closure s | def | dense | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure"
] | A set is dense in a topological space if every point belongs to its closure. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
dense_iff_closure_eq {s : set α} : dense s ↔ closure s = univ | eq_univ_iff_forall.symm | lemma | dense_iff_closure_eq | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"dense"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense.closure_eq {s : set α} (h : dense s) : closure s = univ | dense_iff_closure_eq.mp h | lemma | dense.closure_eq | topology | src/topology/basic.lean | [
"order.filter.ultrafilter",
"algebra.support",
"order.filter.lift"
] | [
"closure",
"dense"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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