Loogle!
Result
Found 74 definitions whose name contains "of_isOpen".
- Dense.inter_of_isOpen_left Mathlib.Topology.Basic
∀ {X : Type u} {s t : Set X} [inst : TopologicalSpace X], Dense s → Dense t → IsOpen s → Dense (s ∩ t) - Dense.inter_of_isOpen_right Mathlib.Topology.Basic
∀ {X : Type u} {s t : Set X} [inst : TopologicalSpace X], Dense s → Dense t → IsOpen t → Dense (s ∩ t) - DenseRange.subset_closure_image_preimage_of_isOpen Mathlib.Topology.Basic
∀ {X : Type u_1} [inst : TopologicalSpace X] {α : Type u_4} {f : α → X} {s : Set X}, DenseRange f → IsOpen s → s ⊆ closure (f '' (f ⁻¹' s)) - TopologicalSpace.generateFrom_setOf_isOpen Mathlib.Topology.Order
∀ {α : Type u} (t : TopologicalSpace α), TopologicalSpace.generateFrom {s | IsOpen s} = t - TopologicalSpace.setOf_isOpen_injective Mathlib.Topology.Order
∀ {α : Type u}, Function.Injective fun t => {s | IsOpen s} - setOf_isOpen_iSup Mathlib.Topology.Order
∀ {α : Type u} {ι : Sort v} {t : ι → TopologicalSpace α}, {s | IsOpen s} = ⋂ i, {s | IsOpen s} - setOf_isOpen_sup Mathlib.Topology.Order
∀ {α : Type u} (t₁ t₂ : TopologicalSpace α), {s | IsOpen s} = {s | IsOpen s} ∩ {s | IsOpen s} - setOf_isOpen_sSup Mathlib.Topology.Order
∀ {α : Type u} {T : Set (TopologicalSpace α)}, {s | IsOpen s} = ⋂ t ∈ T, {s | IsOpen s} - Inducing.setOf_isOpen Mathlib.Topology.Maps
∀ {X : Type u_1} {Y : Type u_2} {f : X → Y} [inst : TopologicalSpace X] [inst_1 : TopologicalSpace Y], Inducing f → {s | IsOpen s} = Set.preimage f '' {t | IsOpen t} - Pairwise.countable_of_isOpen_disjoint Mathlib.Topology.Bases
∀ {α : Type u} [t : TopologicalSpace α] [inst : TopologicalSpace.SeparableSpace α] {ι : Type u_2} {s : ι → Set α}, Pairwise (Disjoint on s) → (∀ (i : ι), IsOpen (s i)) → (∀ (i : ι), Set.Nonempty (s i)) → Countable ι - Set.PairwiseDisjoint.countable_of_isOpen Mathlib.Topology.Bases
∀ {α : Type u} [t : TopologicalSpace α] [inst : TopologicalSpace.SeparableSpace α] {ι : Type u_2} {s : ι → Set α} {a : Set ι}, Set.PairwiseDisjoint a s → (∀ i ∈ a, IsOpen (s i)) → (∀ i ∈ a, Set.Nonempty (s i)) → Set.Countable a - TopologicalSpace.isTopologicalBasis_of_isOpen_of_nhds Mathlib.Topology.Bases
∀ {α : Type u} [t : TopologicalSpace α] {s : Set (Set α)}, (∀ u ∈ s, IsOpen u) → (∀ (a : α) (u : Set α), a ∈ u → IsOpen u → ∃ v ∈ s, a ∈ v ∧ v ⊆ u) → TopologicalSpace.IsTopologicalBasis s - subset_closure_inter_of_isPreirreducible_of_isOpen Mathlib.Topology.Irreducible
∀ {X : Type u_1} [inst : TopologicalSpace X] {S U : Set X}, IsPreirreducible S → IsOpen U → Set.Nonempty (S ∩ U) → S ⊆ closure (S ∩ U) - IsConnected.preimage_of_isOpenMap Mathlib.Topology.Connected.Basic
∀ {α : Type u} {β : Type v} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] {s : Set β}, IsConnected s → ∀ {f : α → β}, Function.Injective f → IsOpenMap f → s ⊆ Set.range f → IsConnected (f ⁻¹' s) - IsPreconnected.preimage_of_isOpenMap Mathlib.Topology.Connected.Basic
∀ {α : Type u} {β : Type v} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] {f : α → β} {s : Set β}, IsPreconnected s → Function.Injective f → IsOpenMap f → s ⊆ Set.range f → IsPreconnected (f ⁻¹' s) - IsCompact.closure_subset_of_isOpen Mathlib.Topology.Separation
∀ {X : Type u_1} [inst : TopologicalSpace X] [inst_1 : R1Space X] {K : Set X}, IsCompact K → ∀ {U : Set X}, IsOpen U → K ⊆ U → closure K ⊆ U - exists_isOpen_singleton_of_isOpen_finite Mathlib.Topology.Separation
∀ {X : Type u_1} [inst : TopologicalSpace X] [inst_1 : T0Space X] {s : Set X}, Set.Finite s → Set.Nonempty s → IsOpen s → ∃ x ∈ s, IsOpen {x} - discreteTopology_of_isOpen_singleton_one Mathlib.Topology.Algebra.Group.Basic
∀ {G : Type w} [inst : TopologicalSpace G] [inst_1 : Group G] [inst_2 : ContinuousMul G], IsOpen {1} → DiscreteTopology G - discreteTopology_of_isOpen_singleton_zero Mathlib.Topology.Algebra.Group.Basic
∀ {G : Type w} [inst : TopologicalSpace G] [inst_1 : AddGroup G] [inst_2 : ContinuousAdd G], IsOpen {0} → DiscreteTopology G - locPathConnected_of_isOpen Mathlib.Topology.Connected.PathConnected
∀ {X : Type u_1} [inst : TopologicalSpace X] [inst_1 : LocPathConnectedSpace X] {U : Set X}, IsOpen U → LocPathConnectedSpace ↑U - isGδ_iInter_of_isOpen Mathlib.Topology.GDelta
∀ {X : Type u_1} {ι' : Sort u_4} [inst : TopologicalSpace X] [inst_1 : Countable ι'] {f : ι' → Set X}, (∀ (i : ι'), IsOpen (f i)) → IsGδ (⋂ i, f i) - IsGδ.iInter_of_isOpen Mathlib.Topology.GDelta
∀ {X : Type u_1} {ι' : Sort u_4} [inst : TopologicalSpace X] [inst_1 : Countable ι'] {f : ι' → Set X}, (∀ (i : ι'), IsOpen (f i)) → IsGδ (⋂ i, f i) - isGδ_biInter_of_isOpen Mathlib.Topology.GDelta
∀ {X : Type u_1} {ι : Type u_3} [inst : TopologicalSpace X] {I : Set ι}, Set.Countable I → ∀ {f : ι → Set X}, (∀ i ∈ I, IsOpen (f i)) → IsGδ (⋂ i ∈ I, f i) - IsGδ.biInter_of_isOpen Mathlib.Topology.GDelta
∀ {X : Type u_1} {ι : Type u_3} [inst : TopologicalSpace X] {I : Set ι}, Set.Countable I → ∀ {f : ι → Set X}, (∀ i ∈ I, IsOpen (f i)) → IsGδ (⋂ i ∈ I, f i) - measurable_of_isOpen Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
∀ {γ : Type u_3} {δ : Type u_5} [inst : TopologicalSpace γ] [inst_1 : MeasurableSpace γ] [inst_2 : BorelSpace γ] [inst_3 : MeasurableSpace δ] {f : δ → γ}, (∀ (s : Set γ), IsOpen s → MeasurableSet (f ⁻¹' s)) → Measurable f - MeasureTheory.Measure.InnerRegularWRT.measurableSet_of_isOpen Mathlib.MeasureTheory.Measure.Regular
∀ {α : Type u_1} [inst : MeasurableSpace α] [inst_1 : TopologicalSpace α] {μ : MeasureTheory.Measure α} {p : Set α → Prop} [inst_2 : MeasureTheory.Measure.OuterRegular μ], MeasureTheory.Measure.InnerRegularWRT μ p IsOpen → (∀ ⦃s U : Set α⦄, p s → IsOpen U → p (s \ U)) → MeasureTheory.Measure.InnerRegularWRT μ p fun s => MeasurableSet s ∧ ↑↑μ s ≠ ⊤ - Convex.strictConvex_of_isOpen Mathlib.Analysis.Convex.Strict
∀ {𝕜 : Type u_1} {E : Type u_3} [inst : OrderedSemiring 𝕜] [inst_1 : TopologicalSpace E] [inst_2 : AddCommMonoid E] [inst_3 : Module 𝕜 E] {s : Set E}, IsOpen s → Convex 𝕜 s → StrictConvex 𝕜 s - gauge_lt_one_of_mem_of_isOpen Mathlib.Analysis.Convex.Gauge
∀ {E : Type u_2} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {s : Set E} [inst_2 : TopologicalSpace E] [inst_3 : ContinuousSMul ℝ E], IsOpen s → ∀ {x : E}, x ∈ s → gauge s x < 1 - gauge_lt_one_eq_self_of_isOpen Mathlib.Analysis.Convex.Gauge
∀ {E : Type u_2} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {s : Set E} [inst_2 : TopologicalSpace E] [inst_3 : ContinuousSMul ℝ E], Convex ℝ s → 0 ∈ s → IsOpen s → {x | gauge s x < 1} = s - gaugeSeminorm_lt_one_of_isOpen Mathlib.Analysis.Convex.Gauge
∀ {𝕜 : Type u_1} {E : Type u_2} [inst : AddCommGroup E] [inst_1 : Module ℝ E] {s : Set E} [inst_2 : RCLike 𝕜] [inst_3 : Module 𝕜 E] [inst_4 : IsScalarTower ℝ 𝕜 E] {hs₀ : Balanced 𝕜 s} {hs₁ : Convex ℝ s} {hs₂ : Absorbent ℝ s} [inst_5 : TopologicalSpace E] [inst_6 : ContinuousSMul ℝ E], IsOpen s → ∀ {x : E}, x ∈ s → (gaugeSeminorm hs₀ hs₁ hs₂) x < 1 - MeasureTheory.Content.outerMeasure_of_isOpen Mathlib.MeasureTheory.Measure.Content
∀ {G : Type w} [inst : TopologicalSpace G] (μ : MeasureTheory.Content G) [inst_1 : R1Space G] (U : Set G) (hU : IsOpen U), ↑(MeasureTheory.Content.outerMeasure μ) U = MeasureTheory.Content.innerContent μ { carrier := U, is_open' := hU } - dense_iInter_of_isOpen_nat Mathlib.Topology.Baire.Lemmas
∀ {X : Type u_1} [inst : TopologicalSpace X] [inst_1 : BaireSpace X] {f : ℕ → Set X}, (∀ (n : ℕ), IsOpen (f n)) → (∀ (n : ℕ), Dense (f n)) → Dense (⋂ n, f n) - dense_iInter_of_isOpen Mathlib.Topology.Baire.Lemmas
∀ {X : Type u_1} {ι : Sort u_3} [inst : TopologicalSpace X] [inst_1 : BaireSpace X] [inst_2 : Countable ι] {f : ι → Set X}, (∀ (i : ι), IsOpen (f i)) → (∀ (i : ι), Dense (f i)) → Dense (⋂ s, f s) - dense_sInter_of_isOpen Mathlib.Topology.Baire.Lemmas
∀ {X : Type u_1} [inst : TopologicalSpace X] [inst_1 : BaireSpace X] {S : Set (Set X)}, (∀ s ∈ S, IsOpen s) → Set.Countable S → (∀ s ∈ S, Dense s) → Dense (⋂₀ S) - dense_biInter_of_isOpen Mathlib.Topology.Baire.Lemmas
∀ {X : Type u_1} {α : Type u_2} [inst : TopologicalSpace X] [inst_1 : BaireSpace X] {S : Set α} {f : α → Set X}, (∀ s ∈ S, IsOpen (f s)) → Set.Countable S → (∀ s ∈ S, Dense (f s)) → Dense (⋂ s ∈ S, f s) - fderivWithin_of_isOpen Mathlib.Analysis.Calculus.FDeriv.Basic
∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f : E → F} {x : E} {s : Set E}, IsOpen s → x ∈ s → fderivWithin 𝕜 f s x = fderiv 𝕜 f x - hasFDerivWithinAt_of_isOpen Mathlib.Analysis.Calculus.FDeriv.Basic
∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] {F : Type u_3} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {f : E → F} {f' : E →L[𝕜] F} {x : E} {s : Set E}, IsOpen s → x ∈ s → (HasFDerivWithinAt f f' s x ↔ HasFDerivAt f f' x) - derivWithin_of_isOpen Mathlib.Analysis.Calculus.Deriv.Basic
∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {F : Type v} [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace 𝕜 F] {f : 𝕜 → F} {x : 𝕜} {s : Set 𝕜}, IsOpen s → x ∈ s → derivWithin f s x = deriv f x - iteratedFDerivWithin_of_isOpen Mathlib.Analysis.Calculus.ContDiff.Defs
∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] {F : Type uF} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {s : Set E} {f : E → F} (n : ℕ), IsOpen s → Set.EqOn (iteratedFDerivWithin 𝕜 n f s) (iteratedFDeriv 𝕜 n f) s - ContDiffOn.continuousOn_fderiv_of_isOpen Mathlib.Analysis.Calculus.ContDiff.Defs
∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] {F : Type uF} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {s : Set E} {f : E → F} {n : ℕ∞}, ContDiffOn 𝕜 n f s → IsOpen s → 1 ≤ n → ContinuousOn (fun x => fderiv 𝕜 f x) s - contDiffOn_top_iff_fderiv_of_isOpen Mathlib.Analysis.Calculus.ContDiff.Defs
∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] {F : Type uF} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {s : Set E} {f : E → F}, IsOpen s → (ContDiffOn 𝕜 ⊤ f s ↔ DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 ⊤ (fun y => fderiv 𝕜 f y) s) - contDiffOn_succ_iff_fderiv_of_isOpen Mathlib.Analysis.Calculus.ContDiff.Defs
∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] {F : Type uF} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {s : Set E} {f : E → F} {n : ℕ}, IsOpen s → (ContDiffOn 𝕜 (↑(n + 1)) f s ↔ DifferentiableOn 𝕜 f s ∧ ContDiffOn 𝕜 (↑n) (fun y => fderiv 𝕜 f y) s) - ContDiffOn.fderiv_of_isOpen Mathlib.Analysis.Calculus.ContDiff.Defs
∀ {𝕜 : Type u} [inst : NontriviallyNormedField 𝕜] {E : Type uE} [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] {F : Type uF} [inst_3 : NormedAddCommGroup F] [inst_4 : NormedSpace 𝕜 F] {s : Set E} {f : E → F} {m n : ℕ∞}, ContDiffOn 𝕜 n f s → IsOpen s → m + 1 ≤ n → ContDiffOn 𝕜 m (fun y => fderiv 𝕜 f y) s - ContDiffOn.continuousOn_deriv_of_isOpen Mathlib.Analysis.Calculus.ContDiff.Basic
∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {F : Type uF} [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace 𝕜 F] {n : ℕ∞} {f₂ : 𝕜 → F} {s₂ : Set 𝕜}, ContDiffOn 𝕜 n f₂ s₂ → IsOpen s₂ → 1 ≤ n → ContinuousOn (deriv f₂) s₂ - contDiffOn_top_iff_deriv_of_isOpen Mathlib.Analysis.Calculus.ContDiff.Basic
∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {F : Type uF} [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace 𝕜 F] {f₂ : 𝕜 → F} {s₂ : Set 𝕜}, IsOpen s₂ → (ContDiffOn 𝕜 ⊤ f₂ s₂ ↔ DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 ⊤ (deriv f₂) s₂) - ContDiffOn.deriv_of_isOpen Mathlib.Analysis.Calculus.ContDiff.Basic
∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {F : Type uF} [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace 𝕜 F] {m n : ℕ∞} {f₂ : 𝕜 → F} {s₂ : Set 𝕜}, ContDiffOn 𝕜 n f₂ s₂ → IsOpen s₂ → m + 1 ≤ n → ContDiffOn 𝕜 m (deriv f₂) s₂ - contDiffOn_succ_iff_deriv_of_isOpen Mathlib.Analysis.Calculus.ContDiff.Basic
∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {F : Type uF} [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace 𝕜 F] {f₂ : 𝕜 → F} {s₂ : Set 𝕜} {n : ℕ}, IsOpen s₂ → (ContDiffOn 𝕜 (↑(n + 1)) f₂ s₂ ↔ DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 (↑n) (deriv f₂) s₂) - WeakSpace.isOpen_of_isOpen Mathlib.Topology.Algebra.Module.WeakDual
∀ {𝕜 : Type u_2} {E : Type u_5} [inst : CommSemiring 𝕜] [inst_1 : TopologicalSpace 𝕜] [inst_2 : ContinuousAdd 𝕜] [inst_3 : ContinuousConstSMul 𝕜 𝕜] [inst_4 : AddCommMonoid E] [inst_5 : Module 𝕜 E] [inst_6 : TopologicalSpace E] (V : Set E), IsOpen (⇑(continuousLinearMapToWeakSpace 𝕜 E) '' V) → IsOpen V - AlgebraicGeometry.Scheme.Hom.isAffineOpen_iff_of_isOpenImmersion Mathlib.AlgebraicGeometry.AffineScheme
∀ {X Y : AlgebraicGeometry.Scheme} (f : AlgebraicGeometry.Scheme.Hom X Y) [H : AlgebraicGeometry.IsOpenImmersion f] {U : TopologicalSpace.Opens ↑↑X.toPresheafedSpace}, AlgebraicGeometry.IsAffineOpen ((AlgebraicGeometry.Scheme.Hom.opensFunctor f).obj U) ↔ AlgebraicGeometry.IsAffineOpen U - AlgebraicGeometry.genericPoint_eq_of_isOpenImmersion Mathlib.AlgebraicGeometry.FunctionField
∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [H : AlgebraicGeometry.IsOpenImmersion f] [hX : IrreducibleSpace ↑↑X.toPresheafedSpace] [inst : IrreducibleSpace ↑↑Y.toPresheafedSpace], f.val.base (genericPoint ↑↑X.toPresheafedSpace) = genericPoint ↑↑Y.toPresheafedSpace - RingHom.PropertyIsLocal.affineLocally_of_isOpenImmersion Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties
∀ {P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop}, RingHom.PropertyIsLocal P → ∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [hf : AlgebraicGeometry.IsOpenImmersion f], AlgebraicGeometry.affineLocally P f - RingHom.PropertyIsLocal.sourceAffineLocally_comp_of_isOpenImmersion Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties
∀ {P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop}, RingHom.PropertyIsLocal P → ∀ {X Y Z : AlgebraicGeometry.Scheme} [inst : AlgebraicGeometry.IsAffine Z] (f : X ⟶ Y) (g : Y ⟶ Z) [inst_1 : AlgebraicGeometry.IsOpenImmersion f], AlgebraicGeometry.sourceAffineLocally P g → AlgebraicGeometry.sourceAffineLocally P (CategoryTheory.CategoryStruct.comp f g) - IsCompact.preimage_of_isOpen Mathlib.Topology.Spectral.Hom
∀ {α : Type u_2} {β : Type u_3} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] {f : α → β} {s : Set β}, IsSpectralMap f → IsCompact s → IsOpen s → IsCompact (f ⁻¹' s) - IsSpectralMap.isCompact_preimage_of_isOpen Mathlib.Topology.Spectral.Hom
∀ {α : Type u_2} {β : Type u_3} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] {f : α → β}, IsSpectralMap f → ∀ ⦃s : Set β⦄, IsOpen s → IsCompact s → IsCompact (f ⁻¹' s) - MeasureTheory.Measure.addHaarScalarFactor_pos_of_isOpenPosMeasure Mathlib.MeasureTheory.Measure.Haar.Unique
∀ {G : Type u_1} [inst : TopologicalSpace G] [inst_1 : AddGroup G] [inst_2 : TopologicalAddGroup G] [inst_3 : MeasurableSpace G] [inst_4 : BorelSpace G] (μ' μ : MeasureTheory.Measure G) [inst_5 : MeasureTheory.Measure.IsAddHaarMeasure μ] [inst_6 : MeasureTheory.Measure.IsAddHaarMeasure μ'], 0 < MeasureTheory.Measure.addHaarScalarFactor μ' μ - MeasureTheory.Measure.haarScalarFactor_pos_of_isOpenPosMeasure Mathlib.MeasureTheory.Measure.Haar.Unique
∀ {G : Type u_1} [inst : TopologicalSpace G] [inst_1 : Group G] [inst_2 : TopologicalGroup G] [inst_3 : MeasurableSpace G] [inst_4 : BorelSpace G] (μ' μ : MeasureTheory.Measure G) [inst_5 : MeasureTheory.Measure.IsHaarMeasure μ] [inst_6 : MeasureTheory.Measure.IsHaarMeasure μ'], 0 < MeasureTheory.Measure.haarScalarFactor μ' μ - MeasureTheory.Measure.measure_isAddHaarMeasure_eq_smul_of_isOpen Mathlib.MeasureTheory.Measure.Haar.Unique
∀ {G : Type u_1} [inst : TopologicalSpace G] [inst_1 : AddGroup G] [inst_2 : TopologicalAddGroup G] [inst_3 : MeasurableSpace G] [inst_4 : BorelSpace G] [inst_5 : LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [inst_6 : MeasureTheory.Measure.IsAddHaarMeasure μ] [inst_7 : MeasureTheory.Measure.IsAddHaarMeasure μ'] {s : Set G}, IsOpen s → ↑↑μ' s = MeasureTheory.Measure.addHaarScalarFactor μ' μ • ↑↑μ s - MeasureTheory.Measure.measure_isHaarMeasure_eq_smul_of_isOpen Mathlib.MeasureTheory.Measure.Haar.Unique
∀ {G : Type u_1} [inst : TopologicalSpace G] [inst_1 : Group G] [inst_2 : TopologicalGroup G] [inst_3 : MeasurableSpace G] [inst_4 : BorelSpace G] [inst_5 : LocallyCompactSpace G] (μ' μ : MeasureTheory.Measure G) [inst_6 : MeasureTheory.Measure.IsHaarMeasure μ] [inst_7 : MeasureTheory.Measure.IsHaarMeasure μ'] {s : Set G}, IsOpen s → ↑↑μ' s = MeasureTheory.Measure.haarScalarFactor μ' μ • ↑↑μ s - lineDerivWithin_of_isOpen Mathlib.Analysis.Calculus.LineDeriv.Basic
∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {F : Type u_2} [inst_1 : NormedAddCommGroup F] [inst_2 : NormedSpace 𝕜 F] {E : Type u_3} [inst_3 : NormedAddCommGroup E] [inst_4 : NormedSpace 𝕜 E] {f : E → F} {s : Set E} {x v : E}, IsOpen s → x ∈ s → lineDerivWithin 𝕜 f s x v = lineDeriv 𝕜 f x v - mfderivWithin_of_isOpen Mathlib.Geometry.Manifold.MFDeriv.Basic
∀ {𝕜 : Type u_1} [inst : NontriviallyNormedField 𝕜] {E : Type u_2} [inst_1 : NormedAddCommGroup E] [inst_2 : NormedSpace 𝕜 E] {H : Type u_3} [inst_3 : TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [inst_4 : TopologicalSpace M] [inst_5 : ChartedSpace H M] {E' : Type u_5} [inst_6 : NormedAddCommGroup E'] [inst_7 : NormedSpace 𝕜 E'] {H' : Type u_6} [inst_8 : TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M' : Type u_7} [inst_9 : TopologicalSpace M'] [inst_10 : ChartedSpace H' M'] {f : M → M'} {x : M} {s : Set M} [Is : SmoothManifoldWithCorners I M] [I's : SmoothManifoldWithCorners I' M'], IsOpen s → x ∈ s → mfderivWithin I I' f s x = mfderiv I I' f x - cardinal_eq_of_isOpen Mathlib.Topology.Algebra.Module.Cardinality
∀ {E : Type u_1} (𝕜 : Type u_2) [inst : NontriviallyNormedField 𝕜] [inst_1 : AddCommGroup E] [inst_2 : Module 𝕜 E] [inst_3 : TopologicalSpace E] [inst_4 : ContinuousAdd E] [inst : ContinuousSMul 𝕜 E] {s : Set E}, IsOpen s → Set.Nonempty s → Cardinal.mk ↑s = Cardinal.mk E - continuum_le_cardinal_of_isOpen Mathlib.Topology.Algebra.Module.Cardinality
∀ {E : Type u_1} (𝕜 : Type u_2) [inst : NontriviallyNormedField 𝕜] [inst_1 : CompleteSpace 𝕜] [inst_2 : AddCommGroup E] [inst_3 : Module 𝕜 E] [inst_4 : Nontrivial E] [inst_5 : TopologicalSpace E] [inst_6 : ContinuousAdd E] [inst : ContinuousSMul 𝕜 E] {s : Set E}, IsOpen s → Set.Nonempty s → Cardinal.continuum ≤ Cardinal.mk ↑s - ContinuousMap.setOfIdeal_ofSet_of_isOpen Mathlib.Topology.ContinuousFunction.Ideals
∀ {X : Type u_1} (𝕜 : Type u_2) [inst : RCLike 𝕜] [inst_1 : TopologicalSpace X] [inst_2 : CompactSpace X] [inst_3 : T2Space X] {s : Set X}, IsOpen s → ContinuousMap.setOfIdeal (ContinuousMap.idealOfSet 𝕜 s) = s - image_subset_closure_compl_image_compl_of_isOpen Mathlib.Topology.ExtremallyDisconnected
∀ {A E : Type u} [inst : TopologicalSpace A] [inst_1 : TopologicalSpace E] {ρ : E → A}, Continuous ρ → Function.Surjective ρ → (∀ (E₀ : Set E), E₀ ≠ Set.univ → IsClosed E₀ → ρ '' E₀ ≠ Set.univ) → ∀ {G : Set E}, IsOpen G → ρ '' G ⊆ closure (ρ '' Gᶜ)ᶜ - eventually_mapsTo_of_isOpen_of_omegaLimit_subset Mathlib.Dynamics.OmegaLimit
∀ {τ : Type u_1} {α : Type u_2} {β : Type u_3} [inst : TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s : Set α) [inst_1 : CompactSpace β] {v : Set β}, IsOpen v → omegaLimit f ϕ s ⊆ v → ∀ᶠ (t : τ) in f, Set.MapsTo (ϕ t) s v - eventually_mapsTo_of_isCompact_absorbing_of_isOpen_of_omegaLimit_subset Mathlib.Dynamics.OmegaLimit
∀ {τ : Type u_1} {α : Type u_2} {β : Type u_3} [inst : TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s : Set α) [inst_1 : T2Space β] {c : Set β}, IsCompact c → (∀ᶠ (t : τ) in f, Set.MapsTo (ϕ t) s c) → ∀ {n : Set β}, IsOpen n → omegaLimit f ϕ s ⊆ n → ∀ᶠ (t : τ) in f, Set.MapsTo (ϕ t) s n - eventually_closure_subset_of_isOpen_of_omegaLimit_subset Mathlib.Dynamics.OmegaLimit
∀ {τ : Type u_1} {α : Type u_2} {β : Type u_3} [inst : TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s : Set α) [inst_1 : CompactSpace β] {v : Set β}, IsOpen v → omegaLimit f ϕ s ⊆ v → ∃ u ∈ f, closure (Set.image2 ϕ u s) ⊆ v - eventually_closure_subset_of_isCompact_absorbing_of_isOpen_of_omegaLimit_subset Mathlib.Dynamics.OmegaLimit
∀ {τ : Type u_1} {α : Type u_2} {β : Type u_3} [inst : TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s : Set α) [inst_1 : T2Space β] {c : Set β}, IsCompact c → (∀ᶠ (t : τ) in f, Set.MapsTo (ϕ t) s c) → ∀ {n : Set β}, IsOpen n → omegaLimit f ϕ s ⊆ n → ∃ u ∈ f, closure (Set.image2 ϕ u s) ⊆ n - eventually_closure_subset_of_isCompact_absorbing_of_isOpen_of_omegaLimit_subset' Mathlib.Dynamics.OmegaLimit
∀ {τ : Type u_1} {α : Type u_2} {β : Type u_3} [inst : TopologicalSpace β] (f : Filter τ) (ϕ : τ → α → β) (s : Set α) {c : Set β}, IsCompact c → (∃ v ∈ f, closure (Set.image2 ϕ v s) ⊆ c) → ∀ {n : Set β}, IsOpen n → omegaLimit f ϕ s ⊆ n → ∃ u ∈ f, closure (Set.image2 ϕ u s) ⊆ n - Ideal.isOpen_of_isOpen_subideal Mathlib.Topology.Algebra.OpenSubgroup
∀ {R : Type u_1} [inst : CommRing R] [inst_1 : TopologicalSpace R] [inst_2 : TopologicalRing R] {U I : Ideal R}, U ≤ I → IsOpen ↑U → IsOpen ↑I - Topology.IsLower.isLowerSet_of_isOpen Mathlib.Topology.Order.LowerUpperTopology
∀ {α : Type u_1} [inst : Preorder α] [inst_1 : TopologicalSpace α] [inst_2 : Topology.IsLower α] {s : Set α}, IsOpen s → IsLowerSet s - Topology.IsUpper.isUpperSet_of_isOpen Mathlib.Topology.Order.LowerUpperTopology
∀ {α : Type u_1} [inst : Preorder α] [inst_1 : TopologicalSpace α] [inst_2 : Topology.IsUpper α] {s : Set α}, IsOpen s → IsUpperSet s - Topology.IsScottHausdorff.dirSupInacc_of_isOpen Mathlib.Topology.Order.ScottTopology
∀ {α : Type u_1} [inst : Preorder α] [inst_1 : TopologicalSpace α] [inst_2 : Topology.IsScottHausdorff α] {s : Set α}, IsOpen s → DirSupInacc s - Topology.IsScott.isUpperSet_of_isOpen Mathlib.Topology.Order.ScottTopology
∀ {α : Type u_1} [inst : Preorder α] [inst_1 : TopologicalSpace α] [inst_2 : Topology.IsScott α] {s : Set α}, IsOpen s → IsUpperSet s
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About
Loogle searches of Lean and Mathlib definitions and theorems.
You may also want to try the CLI version, the VS
Code extension, the lean.nvim
integration or the Zulip bot.
Usage
Loogle finds definitions and lemmas in various ways:
By constant:
🔍Real.sin
finds all lemmas whose statement somehow mentions the sine function.By lemma name substring:
🔍"differ"
finds all lemmas that have"differ"
somewhere in their lemma name.By subexpression:
🔍_ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.The pattern can also be non-linear, as in
🔍Real.sqrt ?a * Real.sqrt ?a
If the pattern has parameters, they are matched in any order. Both of these will find
List.map
:
🔍(?a -> ?b) -> List ?a -> List ?b
🔍List ?a -> (?a -> ?b) -> List ?b
By main conclusion:
🔍|- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all→
and∀
) has the given shape.As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍|- _ < _ → tsum _ < tsum _
will findtsum_lt_tsum
even though the hypothesisf i < g i
is not the last.
If you pass more than one such search filter, separated by commas
Loogle will return lemmas which match all of them. The
search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
woould find all lemmas which mention the constants Real.sin
and tsum
, have "two"
as a substring of the
lemma name, include a product and a power somewhere in the type,
and have a hypothesis of the form _ < _
(if
there were any such lemmas). Metavariables (?a
) are
assigned independently in each filter.
The #lucky
button will directly send you to the
documentation of the first hit.
Source code
You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is currently provided by Joachim Breitner <mail@joachim-breitner.de>.
This is Loogle revision 34713b2
serving mathlib revision a547a94