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values | symbolic_name stringlengths 1 131 | library stringclasses 417
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coe_bot : (↑(⊥ : compacts α) : set α) = ∅ | rfl | lemma | topological_space.compacts.coe_bot | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_finset_sup {ι : Type*} {s : finset ι} {f : ι → compacts α} :
(↑(s.sup f) : set α) = s.sup (λ i, f i) | begin
classical,
refine finset.induction_on s rfl (λ a s _ h, _),
simp_rw [finset.sup_insert, coe_sup, sup_eq_union],
congr',
end | lemma | topological_space.compacts.coe_finset_sup | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"finset",
"finset.induction_on",
"finset.sup_insert"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : α → β) (hf : continuous f) (K : compacts α) : compacts β | ⟨f '' K.1, K.2.image hf⟩ | def | topological_space.compacts.map | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"continuous"
] | The image of a compact set under a continuous function. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_map {f : α → β} (hf : continuous f) (s : compacts α) :
(s.map f hf : set β) = f '' s | rfl | lemma | topological_space.compacts.coe_map | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"continuous"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id (K : compacts α) : K.map id continuous_id = K | compacts.ext $ set.image_id _ | lemma | topological_space.compacts.map_id | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"continuous_id",
"map_id",
"set.image_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp (f : β → γ) (g : α → β) (hf : continuous f) (hg : continuous g) (K : compacts α) :
K.map (f ∘ g) (hf.comp hg) = (K.map g hg).map f hf | compacts.ext $ set.image_comp _ _ _ | lemma | topological_space.compacts.map_comp | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"continuous",
"map_comp",
"set.image_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv (f : α ≃ₜ β) : compacts α ≃ compacts β | { to_fun := compacts.map f f.continuous,
inv_fun := compacts.map _ f.symm.continuous,
left_inv := λ s, by { ext1, simp only [coe_map, ← image_comp, f.symm_comp_self, image_id] },
right_inv := λ s, by { ext1, simp only [coe_map, ← image_comp, f.self_comp_symm, image_id] } } | def | topological_space.compacts.equiv | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"equiv",
"inv_fun"
] | A homeomorphism induces an equivalence on compact sets, by taking the image. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
equiv_refl : compacts.equiv (homeomorph.refl α) = equiv.refl _ | equiv.ext map_id | lemma | topological_space.compacts.equiv_refl | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"equiv.ext",
"equiv.refl",
"homeomorph.refl",
"map_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_trans (f : α ≃ₜ β) (g : β ≃ₜ γ) :
compacts.equiv (f.trans g) = (compacts.equiv f).trans (compacts.equiv g) | equiv.ext $ map_comp _ _ _ _ | lemma | topological_space.compacts.equiv_trans | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"equiv.ext",
"map_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
equiv_symm (f : α ≃ₜ β) : compacts.equiv f.symm = (compacts.equiv f).symm | rfl | lemma | topological_space.compacts.equiv_symm | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_equiv_apply_eq_preimage (f : α ≃ₜ β) (K : compacts α) :
(compacts.equiv f K : set β) = f.symm ⁻¹' (K : set α) | f.to_equiv.image_eq_preimage K | lemma | topological_space.compacts.coe_equiv_apply_eq_preimage | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [] | The image of a compact set under a homeomorphism can also be expressed as a preimage. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod (K : compacts α) (L : compacts β) : compacts (α × β) | { carrier := K ×ˢ L,
is_compact' := is_compact.prod K.2 L.2 } | def | topological_space.compacts.prod | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"is_compact.prod"
] | The product of two `compacts`, as a `compacts` in the product space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod (K : compacts α) (L : compacts β) : (K.prod L : set (α × β)) = K ×ˢ L | rfl | lemma | topological_space.compacts.coe_prod | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_compacts (α : Type*) [topological_space α] extends compacts α | (nonempty' : carrier.nonempty) | structure | topological_space.nonempty_compacts | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"topological_space"
] | The type of nonempty compact sets of a topological space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact (s : nonempty_compacts α) : is_compact (s : set α) | s.is_compact' | lemma | topological_space.nonempty_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 | |
nonempty (s : nonempty_compacts α) : (s : set α).nonempty | s.nonempty' | lemma | topological_space.nonempty_compacts.nonempty | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_closeds [t2_space α] (s : nonempty_compacts α) : closeds α | ⟨s, s.is_compact.is_closed⟩ | def | topological_space.nonempty_compacts.to_closeds | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"t2_space"
] | Reinterpret a nonempty compact as a closed set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {s t : nonempty_compacts α} (h : (s : set α) = t) : s = t | set_like.ext' h | lemma | topological_space.nonempty_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 | |
coe_mk (s : compacts α) (h) : (mk s h : set α) = s | rfl | lemma | topological_space.nonempty_compacts.coe_mk | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
carrier_eq_coe (s : nonempty_compacts α) : s.carrier = s | rfl | lemma | topological_space.nonempty_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 : nonempty_compacts α) : (↑(s ⊔ t) : set α) = s ∪ t | rfl | lemma | topological_space.nonempty_compacts.coe_sup | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_top [compact_space α] [nonempty α] :
(↑(⊤ : nonempty_compacts α) : set α) = univ | rfl | lemma | topological_space.nonempty_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 | |
to_compact_space {s : nonempty_compacts α} : compact_space s | is_compact_iff_compact_space.1 s.is_compact | instance | topological_space.nonempty_compacts.to_compact_space | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nonempty {s : nonempty_compacts α} : nonempty s | s.nonempty.to_subtype | instance | topological_space.nonempty_compacts.to_nonempty | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod (K : nonempty_compacts α) (L : nonempty_compacts β) :
nonempty_compacts (α × β) | { nonempty' := K.nonempty.prod L.nonempty,
.. K.to_compacts.prod L.to_compacts } | def | topological_space.nonempty_compacts.prod | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [] | The product of two `nonempty_compacts`, as a `nonempty_compacts` in the product space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod (K : nonempty_compacts α) (L : nonempty_compacts β) :
(K.prod L : set (α × β)) = K ×ˢ L | rfl | lemma | topological_space.nonempty_compacts.coe_prod | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
positive_compacts (α : Type*) [topological_space α] extends compacts α | (interior_nonempty' : (interior carrier).nonempty) | structure | topological_space.positive_compacts | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"interior",
"topological_space"
] | The type of compact sets with nonempty interior of a topological space.
See also `compacts` and `nonempty_compacts`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact (s : positive_compacts α) : is_compact (s : set α) | s.is_compact' | lemma | topological_space.positive_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 | |
interior_nonempty (s : positive_compacts α) : (interior (s : set α)).nonempty | s.interior_nonempty' | lemma | topological_space.positive_compacts.interior_nonempty | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"interior"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty (s : positive_compacts α) : (s : set α).nonempty | s.interior_nonempty.mono interior_subset | lemma | topological_space.positive_compacts.nonempty | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"interior_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_nonempty_compacts (s : positive_compacts α) : nonempty_compacts α | ⟨s.to_compacts, s.nonempty⟩ | def | topological_space.positive_compacts.to_nonempty_compacts | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [] | Reinterpret a positive compact as a nonempty compact. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {s t : positive_compacts α} (h : (s : set α) = t) : s = t | set_like.ext' h | lemma | topological_space.positive_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 : positive_compacts α) : s.carrier = s | rfl | lemma | topological_space.positive_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 : positive_compacts α) : (↑(s ⊔ t) : set α) = s ∪ t | rfl | lemma | topological_space.positive_compacts.coe_sup | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_top [compact_space α] [nonempty α] :
(↑(⊤ : positive_compacts α) : set α) = univ | rfl | lemma | topological_space.positive_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 | |
map (f : α → β) (hf : continuous f) (hf' : is_open_map f) (K : positive_compacts α) :
positive_compacts β | { interior_nonempty' :=
(K.interior_nonempty'.image _).mono (hf'.image_interior_subset K.to_compacts),
..K.map f hf } | def | topological_space.positive_compacts.map | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"continuous",
"is_open_map"
] | The image of a positive compact set under a continuous open map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_map {f : α → β} (hf : continuous f) (hf' : is_open_map f)
(s : positive_compacts α) :
(s.map f hf hf' : set β) = f '' s | rfl | lemma | topological_space.positive_compacts.coe_map | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"continuous",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id (K : positive_compacts α) : K.map id continuous_id is_open_map.id = K | positive_compacts.ext $ set.image_id _ | lemma | topological_space.positive_compacts.map_id | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"continuous_id",
"is_open_map.id",
"map_id",
"set.image_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp (f : β → γ) (g : α → β) (hf : continuous f) (hg : continuous g)
(hf' : is_open_map f) (hg' : is_open_map g)
(K : positive_compacts α) :
K.map (f ∘ g) (hf.comp hg) (hf'.comp hg') = (K.map g hg hg').map f hf hf' | positive_compacts.ext $ set.image_comp _ _ _ | lemma | topological_space.positive_compacts.map_comp | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"continuous",
"is_open_map",
"map_comp",
"set.image_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
_root_.exists_positive_compacts_subset [locally_compact_space α] {U : set α} (ho : is_open U)
(hn : U.nonempty) : ∃ K : positive_compacts α, ↑K ⊆ U | let ⟨x, hx⟩ := hn, ⟨K, hKc, hxK, hKU⟩ := exists_compact_subset ho hx in ⟨⟨⟨K, hKc⟩, ⟨x, hxK⟩⟩, hKU⟩ | lemma | exists_positive_compacts_subset | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"exists_compact_subset",
"is_open",
"locally_compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty' [locally_compact_space α] [nonempty α] : nonempty (positive_compacts α) | nonempty_of_exists $ exists_positive_compacts_subset is_open_univ univ_nonempty | instance | topological_space.positive_compacts.nonempty' | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"exists_positive_compacts_subset",
"is_open_univ",
"locally_compact_space"
] | In a nonempty locally compact space, there exists a compact set with nonempty interior. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
prod (K : positive_compacts α) (L : positive_compacts β) :
positive_compacts (α × β) | { interior_nonempty' :=
begin
simp only [compacts.carrier_eq_coe, compacts.coe_prod, interior_prod_eq],
exact K.interior_nonempty.prod L.interior_nonempty,
end,
.. K.to_compacts.prod L.to_compacts } | def | topological_space.positive_compacts.prod | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"interior_prod_eq"
] | The product of two `positive_compacts`, as a `positive_compacts` in the product space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod (K : positive_compacts α) (L : positive_compacts β) :
(K.prod L : set (α × β)) = K ×ˢ L | rfl | lemma | topological_space.positive_compacts.coe_prod | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
compact_opens (α : Type*) [topological_space α] extends compacts α | (is_open' : is_open carrier) | structure | topological_space.compact_opens | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"is_open",
"topological_space"
] | The type of compact open sets of a topological space. This is useful in non Hausdorff contexts,
in particular spectral spaces. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact (s : compact_opens α) : is_compact (s : set α) | s.is_compact' | lemma | topological_space.compact_opens.is_compact | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"is_compact"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open (s : compact_opens α) : is_open (s : set α) | s.is_open' | lemma | topological_space.compact_opens.is_open | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
to_opens (s : compact_opens α) : opens α | ⟨s, s.is_open⟩ | def | topological_space.compact_opens.to_opens | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [] | Reinterpret a compact open as an open. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
to_clopens [t2_space α] (s : compact_opens α) : clopens α | ⟨s, s.is_open, s.is_compact.is_closed⟩ | def | topological_space.compact_opens.to_clopens | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"t2_space"
] | Reinterpret a compact open as a clopen. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
ext {s t : compact_opens α} (h : (s : set α) = t) : s = t | set_like.ext' h | lemma | topological_space.compact_opens.ext | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"set_like.ext'"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sup (s t : compact_opens α) : (↑(s ⊔ t) : set α) = s ∪ t | rfl | lemma | topological_space.compact_opens.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 : compact_opens α) : (↑(s ⊓ t) : set α) = s ∩ t | rfl | lemma | topological_space.compact_opens.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] : (↑(⊤ : compact_opens α) : set α) = univ | rfl | lemma | topological_space.compact_opens.coe_top | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"compact_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bot : (↑(⊥ : compact_opens α) : set α) = ∅ | rfl | lemma | topological_space.compact_opens.coe_bot | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sdiff [t2_space α] (s t : compact_opens α) : (↑(s \ t) : set α) = s \ t | rfl | lemma | topological_space.compact_opens.coe_sdiff | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_compl [t2_space α] [compact_space α] (s : compact_opens α) : (↑sᶜ : set α) = sᶜ | rfl | lemma | topological_space.compact_opens.coe_compl | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"compact_space",
"t2_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map (f : α → β) (hf : continuous f) (hf' : is_open_map f) (s : compact_opens α) :
compact_opens β | ⟨s.to_compacts.map f hf, hf' _ s.is_open⟩ | def | topological_space.compact_opens.map | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"continuous",
"is_open_map"
] | The image of a compact open under a continuous open map. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_map {f : α → β} (hf : continuous f) (hf' : is_open_map f)
(s : compact_opens α) : (s.map f hf hf' : set β) = f '' s | rfl | lemma | topological_space.compact_opens.coe_map | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"continuous",
"is_open_map"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_id (K : compact_opens α) : K.map id continuous_id is_open_map.id = K | compact_opens.ext $ set.image_id _ | lemma | topological_space.compact_opens.map_id | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"continuous_id",
"is_open_map.id",
"map_id",
"set.image_id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
map_comp (f : β → γ) (g : α → β) (hf : continuous f) (hg : continuous g)
(hf' : is_open_map f) (hg' : is_open_map g)
(K : compact_opens α) :
K.map (f ∘ g) (hf.comp hg) (hf'.comp hg') = (K.map g hg hg').map f hf hf' | compact_opens.ext $ set.image_comp _ _ _ | lemma | topological_space.compact_opens.map_comp | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [
"continuous",
"is_open_map",
"map_comp",
"set.image_comp"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod (K : compact_opens α) (L : compact_opens β) :
compact_opens (α × β) | { is_open' := K.is_open.prod L.is_open,
.. K.to_compacts.prod L.to_compacts } | def | topological_space.compact_opens.prod | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [] | The product of two `compact_opens`, as a `compact_opens` in the product space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
coe_prod (K : compact_opens α) (L : compact_opens β) :
(K.prod L : set (α × β)) = K ×ˢ L | rfl | lemma | topological_space.compact_opens.coe_prod | topology.sets | src/topology/sets/compacts.lean | [
"topology.sets.closeds",
"topology.quasi_separated"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
opens | (carrier : set α)
(is_open' : is_open carrier) | structure | topological_space.opens | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"is_open"
] | The type of open subsets of a topological space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
«forall» {p : opens α → Prop} : (∀ U, p U) ↔ ∀ (U : set α) (hU : is_open U), p ⟨U, hU⟩ | ⟨λ h _ _, h _, λ h ⟨U, hU⟩, h _ _⟩ | lemma | topological_space.opens.«forall» | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
carrier_eq_coe (U : opens α) : U.1 = ↑U | rfl | lemma | topological_space.opens.carrier_eq_coe | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_mk {U : set α} {hU : is_open U} : ↑(⟨U, hU⟩ : opens α) = U | rfl | lemma | topological_space.opens.coe_mk | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"is_open"
] | the coercion `opens α → set α` applied to a pair is the same as taking the first component | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mem_mk {x : α} {U : set α} {h : is_open U} :
@has_mem.mem _ (opens α) _ x ⟨U, h⟩ ↔ x ∈ U | iff.rfl | lemma | topological_space.opens.mem_mk | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_coe_sort {U : opens α} : nonempty U ↔ (U : set α).nonempty | set.nonempty_coe_sort | lemma | topological_space.opens.nonempty_coe_sort | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"set.nonempty_coe_sort"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ext {U V : opens α} (h : (U : set α) = V) : U = V | set_like.coe_injective h | lemma | topological_space.opens.ext | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"set_like.coe_injective"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inj {U V : opens α} : (U : set α) = V ↔ U = V | set_like.ext'_iff.symm | lemma | topological_space.opens.coe_inj | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open (U : opens α) : is_open (U : set α) | U.is_open' | lemma | topological_space.opens.is_open | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"is_open"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mk_coe (U : opens α) : mk ↑U U.is_open = U | by { cases U, refl } | lemma | topological_space.opens.mk_coe | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
simps.coe (U : opens α) : set α | U
initialize_simps_projections opens (carrier → coe) | def | topological_space.opens.simps.coe | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [] | See Note [custom simps projection]. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
interior (s : set α) : opens α | ⟨interior s, is_open_interior⟩ | def | topological_space.opens.interior | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"interior"
] | The interior of a set, as an element of `opens`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
gc : galois_connection (coe : opens α → set α) interior | λ U s, ⟨λ h, interior_maximal h U.is_open, λ h, le_trans h interior_subset⟩ | lemma | topological_space.opens.gc | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"galois_connection",
"interior",
"interior_maximal"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
gi : galois_coinsertion coe (@interior α _) | { choice := λ s hs, ⟨s, interior_eq_iff_is_open.mp $ le_antisymm interior_subset hs⟩,
gc := gc,
u_l_le := λ _, interior_subset,
choice_eq := λ s hs, le_antisymm hs interior_subset } | def | topological_space.opens.gi | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"galois_coinsertion",
"interior",
"interior_subset"
] | The galois coinsertion between sets and opens. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
mk_inf_mk {U V : set α} {hU : is_open U} {hV : is_open V} :
(⟨U, hU⟩ ⊓ ⟨V, hV⟩ : opens α) = ⟨U ⊓ V, is_open.inter hU hV⟩ | rfl | lemma | topological_space.opens.mk_inf_mk | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"is_open",
"is_open.inter"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_inf (s t : opens α) : (↑(s ⊓ t) : set α) = s ∩ t | rfl | lemma | topological_space.opens.coe_inf | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_sup (s t : opens α) : (↑(s ⊔ t) : set α) = s ∪ t | rfl | lemma | topological_space.opens.coe_sup | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_bot : ((⊥ : opens α) : set α) = ∅ | rfl | lemma | topological_space.opens.coe_bot | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_top : ((⊤ : opens α) : set α) = set.univ | rfl | lemma | topological_space.opens.coe_top | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_Sup {S : set (opens α)} : (↑(Sup S) : set α) = ⋃ i ∈ S, ↑i | rfl | lemma | topological_space.opens.coe_Sup | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_finset_sup (f : ι → opens α) (s : finset ι) :
(↑(s.sup f) : set α) = s.sup (coe ∘ f) | map_finset_sup (⟨⟨coe, coe_sup⟩, coe_bot⟩ : sup_bot_hom (opens α) (set α)) _ _ | lemma | topological_space.opens.coe_finset_sup | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"finset",
"map_finset_sup",
"sup_bot_hom"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_finset_inf (f : ι → opens α) (s : finset ι) :
(↑(s.inf f) : set α) = s.inf (coe ∘ f) | map_finset_inf (⟨⟨coe, coe_inf⟩, coe_top⟩ : inf_top_hom (opens α) (set α)) _ _ | lemma | topological_space.opens.coe_finset_inf | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"finset",
"inf_top_hom",
"map_finset_inf"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_def {ι} (s : ι → opens α) : (⨆ i, s i) = ⟨⋃ i, s i, is_open_Union $ λ i, (s i).2⟩ | by { ext, simp only [supr, coe_Sup, bUnion_range], refl } | lemma | topological_space.opens.supr_def | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"is_open_Union",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
supr_mk {ι} (s : ι → set α) (h : Π i, is_open (s i)) :
(⨆ i, ⟨s i, h i⟩ : opens α) = ⟨⋃ i, s i, is_open_Union h⟩ | by { rw supr_def, simp } | lemma | topological_space.opens.supr_mk | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"is_open",
"is_open_Union"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
coe_supr {ι} (s : ι → opens α) :
((⨆ i, s i : opens α) : set α) = ⋃ i, s i | by simp [supr_def] | lemma | topological_space.opens.coe_supr | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_supr {ι} {x : α} {s : ι → opens α} : x ∈ supr s ↔ ∃ i, x ∈ s i | by { rw [← set_like.mem_coe], simp, } | theorem | topological_space.opens.mem_supr | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"set_like.mem_coe",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_Sup {Us : set (opens α)} {x : α} : x ∈ Sup Us ↔ ∃ u ∈ Us, x ∈ u | by simp_rw [Sup_eq_supr, mem_supr] | lemma | topological_space.opens.mem_Sup | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"Sup_eq_supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
open_embedding_of_le {U V : opens α} (i : U ≤ V) :
open_embedding (set.inclusion i) | { inj := set.inclusion_injective i,
induced := (@induced_compose _ _ _ _ (set.inclusion i) coe).symm,
open_range :=
begin
rw set.range_inclusion i,
exact U.is_open.preimage continuous_subtype_val
end, } | lemma | topological_space.opens.open_embedding_of_le | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"continuous_subtype_val",
"induced_compose",
"open_embedding",
"set.inclusion",
"set.inclusion_injective",
"set.range_inclusion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
not_nonempty_iff_eq_bot (U : opens α) : ¬ set.nonempty (U : set α) ↔ U = ⊥ | by rw [← coe_inj, opens.coe_bot, ← set.not_nonempty_iff_eq_empty] | lemma | topological_space.opens.not_nonempty_iff_eq_bot | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"set.nonempty",
"set.not_nonempty_iff_eq_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
ne_bot_iff_nonempty (U : opens α) : U ≠ ⊥ ↔ set.nonempty (U : set α) | by rw [ne.def, ← opens.not_nonempty_iff_eq_bot, not_not] | lemma | topological_space.opens.ne_bot_iff_nonempty | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"not_not",
"set.nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
eq_bot_or_top {α} [t : topological_space α] (h : t = ⊤) (U : opens α) : U = ⊥ ∨ U = ⊤ | begin
simp only [← coe_inj],
unfreezingI { subst h }, letI : topological_space α := ⊤,
exact (is_open_top_iff _).1 U.2
end | lemma | topological_space.opens.eq_bot_or_top | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"topological_space"
] | An open set in the indiscrete topology is either empty or the whole space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_basis (B : set (opens α)) : Prop | is_topological_basis ((coe : _ → set α) '' B) | def | topological_space.opens.is_basis | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [] | A set of `opens α` is a basis if the set of corresponding sets is a topological basis. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_basis_iff_nbhd {B : set (opens α)} :
is_basis B ↔ ∀ {U : opens α} {x}, x ∈ U → ∃ U' ∈ B, x ∈ U' ∧ U' ≤ U | begin
split; intro h,
{ rintros ⟨sU, hU⟩ x hx,
rcases h.mem_nhds_iff.mp (is_open.mem_nhds hU hx)
with ⟨sV, ⟨⟨V, H₁, H₂⟩, hsV⟩⟩,
refine ⟨V, H₁, _⟩,
cases V, dsimp at H₂, subst H₂, exact hsV },
{ refine is_topological_basis_of_open_of_nhds _ _,
{ rintros sU ⟨U, ⟨H₁, rfl⟩⟩, exact U.2 },
{ i... | lemma | topological_space.opens.is_basis_iff_nbhd | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"is_open.mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_basis_iff_cover {B : set (opens α)} :
is_basis B ↔ ∀ U : opens α, ∃ Us ⊆ B, U = Sup Us | begin
split,
{ intros hB U,
refine ⟨{V : opens α | V ∈ B ∧ V ≤ U}, λ U hU, hU.left, _⟩,
apply ext,
rw [coe_Sup, hB.open_eq_sUnion' U.is_open],
simp_rw [sUnion_eq_bUnion, Union, supr_and, supr_image],
refl },
{ intro h,
rw is_basis_iff_nbhd,
intros U x hx,
rcases h U with ⟨Us, hUs, ... | lemma | topological_space.opens.is_basis_iff_cover | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"le_Sup",
"supr_and",
"supr_image"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_basis.is_compact_open_iff_eq_finite_Union
{ι : Type*} (b : ι → opens α) (hb : is_basis (set.range b))
(hb' : ∀ i, is_compact (b i : set α)) (U : set α) :
is_compact U ∧ is_open U ↔ ∃ (s : set ι), s.finite ∧ U = ⋃ i ∈ s, b i | begin
apply is_compact_open_iff_eq_finite_Union_of_is_topological_basis
(λ i : ι, (b i).1),
{ convert hb, ext, simp },
{ exact hb' }
end | lemma | topological_space.opens.is_basis.is_compact_open_iff_eq_finite_Union | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"is_compact",
"is_compact_open_iff_eq_finite_Union_of_is_topological_basis",
"is_open",
"set.range"
] | If `α` has a basis consisting of compact opens, then an open set in `α` is compact open iff
it is a finite union of some elements in the basis | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_compact_element_iff (s : opens α) :
complete_lattice.is_compact_element s ↔ is_compact (s : set α) | begin
rw [is_compact_iff_finite_subcover, complete_lattice.is_compact_element_iff],
refine ⟨_, λ H ι U hU, _⟩,
{ introv H hU hU',
obtain ⟨t, ht⟩ := H ι (λ i, ⟨U i, hU i⟩) (by simpa),
refine ⟨t, set.subset.trans ht _⟩,
rw [coe_finset_sup, finset.sup_eq_supr],
refl },
{ obtain ⟨t, ht⟩ := H (λ i, U... | lemma | topological_space.opens.is_compact_element_iff | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"complete_lattice.is_compact_element",
"complete_lattice.is_compact_element_iff",
"finset.le_sup",
"finset.sup_eq_supr",
"is_compact",
"is_compact_iff_finite_subcover",
"is_open",
"set.Union_subset_iff",
"set.subset.trans",
"supr"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap (f : C(α, β)) : frame_hom (opens β) (opens α) | { to_fun := λ s, ⟨f ⁻¹' s, s.2.preimage f.continuous⟩,
map_Sup' := λ s, ext $ by simp only [coe_Sup, preimage_Union, bUnion_image, coe_mk],
map_inf' := λ a b, rfl,
map_top' := rfl } | def | topological_space.opens.comap | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"frame_hom"
] | The preimage of an open set, as an open set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
comap_id : comap (continuous_map.id α) = frame_hom.id _ | frame_hom.ext $ λ a, ext rfl | lemma | topological_space.opens.comap_id | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"continuous_map.id",
"frame_hom.ext",
"frame_hom.id"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
comap_mono (f : C(α, β)) {s t : opens β} (h : s ≤ t) : comap f s ≤ comap f t | order_hom_class.mono (comap f) h | lemma | topological_space.opens.comap_mono | topology.sets | src/topology/sets/opens.lean | [
"order.hom.complete_lattice",
"topology.bases",
"topology.homeomorph",
"topology.continuous_function.basic",
"order.compactly_generated",
"tactic.auto_cases"
] | [
"order_hom_class.mono"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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