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Result

Found 125 declarations mentioning LocallyFinite.

• LocallyFinite 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] (f : ι → Set X) : Prop
• locallyFinite_of_finite 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] [Finite ι] (f : ι → Set X) : LocallyFinite f
• LocallyFinite.closure 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} (hf : LocallyFinite f) : LocallyFinite fun i => closure (f i)
• LocallyFinite.option_elim' 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} (hf : LocallyFinite f) (s : Set X) : LocallyFinite (Option.elim' s f)
• LocallyFinite.isClosed_iUnion 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} (hf : LocallyFinite f) (hc : ∀ (i : ι), IsClosed (f i)) : IsClosed (⋃ i, f i)
• LocallyFinite.point_finite 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} (hf : LocallyFinite f) (x : X) : {b | x ∈ f b}.Finite
• locallyFinite_option 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f : Option ι → Set X} : LocallyFinite f ↔ LocallyFinite (f ∘ some)
• LocallyFinite.comp_injective 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {ι' : Type u_2} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} {g : ι' → ι} (hf : LocallyFinite f) (hg : Function.Injective g) : LocallyFinite (f ∘ g)
• LocallyFinite.of_comp_surjective 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {ι' : Type u_2} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} {g : ι' → ι} (hg : Function.Surjective g) (hfg : LocallyFinite (f ∘ g)) : LocallyFinite f
• LocallyFinite.subset 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f g : ι → Set X} (hf : LocallyFinite f) (hg : ∀ (i : ι), g i ⊆ f i) : LocallyFinite g
• LocallyFinite.closure_iUnion 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} (h : LocallyFinite f) : closure (⋃ i, f i) = ⋃ i, closure (f i)
• LocallyFinite.preimage_continuous 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] {f : ι → Set X} {g : Y → X} (hf : LocallyFinite f) (hg : Continuous g) : LocallyFinite fun x => g ⁻¹' f x
• LocallyFinite.sumElim 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {ι' : Type u_2} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} {g : ι' → Set X} (hf : LocallyFinite f) (hg : LocallyFinite g) : LocallyFinite (Sum.elim f g)
• LocallyFinite.comp_injOn 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {ι' : Type u_2} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} {g : ι' → ι} (hf : LocallyFinite f) (hg : Set.InjOn g {i | (f (g i)).Nonempty}) : LocallyFinite (f ∘ g)
• LocallyFinite.eventually_smallSets 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} (hf : LocallyFinite f) (x : X) : ∀ᶠ (s : Set X) in (nhds x).smallSets, {i | (f i ∩ s).Nonempty}.Finite
• LocallyFinite.nhdsWithin_iUnion 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} (hf : LocallyFinite f) (a : X) : nhdsWithin a (⋃ i, f i) = ⨆ i, nhdsWithin a (f i)
• locallyFinite_iff_smallSets 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} : LocallyFinite f ↔ ∀ (x : X), ∀ᶠ (s : Set X) in (nhds x).smallSets, {i | (f i ∩ s).Nonempty}.Finite
• Equiv.locallyFinite_comp_iff 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {ι' : Type u_2} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} (e : ι' ≃ ι) : LocallyFinite (f ∘ ⇑e) ↔ LocallyFinite f
• LocallyFinite.continuousOn_iUnion 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] {f : ι → Set X} {g : X → Y} (hf : LocallyFinite f) (h_cl : ∀ (i : ι), IsClosed (f i)) (h_cont : ∀ (i : ι), ContinuousOn g (f i)) : ContinuousOn g (⋃ i, f i)
• LocallyFinite.prod_left 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] {g : ι → Set Y} (hg : LocallyFinite g) (f : ι → Set X) : LocallyFinite fun i => f i ×ˢ g i
• LocallyFinite.prod_right 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] {f : ι → Set X} (hf : LocallyFinite f) (g : ι → Set Y) : LocallyFinite fun i => f i ×ˢ g i
• locallyFinite_sum 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {ι' : Type u_2} {X : Type u_4} [TopologicalSpace X] {f : ι ⊕ ι' → Set X} : LocallyFinite f ↔ LocallyFinite (f ∘ Sum.inl) ∧ LocallyFinite (f ∘ Sum.inr)
• LocallyFinite.continuous 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] {f : ι → Set X} {g : X → Y} (hf : LocallyFinite f) (h_cov : ⋃ i, f i = Set.univ) (h_cl : ∀ (i : ι), IsClosed (f i)) (h_cont : ∀ (i : ι), ContinuousOn g (f i)) : Continuous g
• LocallyFinite.exists_mem_basis 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} {ι' : Sort u_6} (hf : LocallyFinite f) {p : ι' → Prop} {s : ι' → Set X} {x : X} (hb : (nhds x).HasBasis p s) : ∃ i, p i ∧ {j | (f j ∩ s i).Nonempty}.Finite
• LocallyFinite.continuousOn_iUnion' 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] {f : ι → Set X} {g : X → Y} (hf : LocallyFinite f) (hc : ∀ (i : ι), ∀ x ∈ closure (f i), ContinuousWithinAt g (f i) x) : ContinuousOn g (⋃ i, f i)
• LocallyFinite.eventually_subset 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {s : ι → Set X} (hs : LocallyFinite s) (hs' : ∀ (i : ι), IsClosed (s i)) (x : X) : ∀ᶠ (y : X) in nhds x, {i | y ∈ s i} ⊆ {i | x ∈ s i}
• LocallyFinite.on_range 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} (hf : LocallyFinite f) : LocallyFinite Subtype.val
• LocallyFinite.continuous' 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} {Y : Type u_5} [TopologicalSpace X] [TopologicalSpace Y] {f : ι → Set X} {g : X → Y} (hf : LocallyFinite f) (h_cov : ⋃ i, f i = Set.univ) (hc : ∀ (i : ι), ∀ x ∈ closure (f i), ContinuousWithinAt g (f i) x) : Continuous g
• LocallyFinite.exists_forall_eventually_atTop_eventuallyEq 📋 Mathlib.Topology.LocallyFinite
  {α : Type u_3} {X : Type u_4} [TopologicalSpace X] {f : ℕ → X → α} (hf : LocallyFinite fun n => {x | f (n + 1) x ≠ f n x}) : ∃ F, ∀ (x : X), ∀ᶠ (n : ℕ) in Filter.atTop, f n =ᶠ[nhds x] F
• LocallyFinite.iInter_compl_mem_nhds 📋 Mathlib.Topology.LocallyFinite
  {ι : Type u_1} {X : Type u_4} [TopologicalSpace X] {f : ι → Set X} (hf : LocallyFinite f) (hc : ∀ (i : ι), IsClosed (f i)) (x : X) : ⋂ i, ⋂ (_ : x ∉ f i), (f i)ᶜ ∈ nhds x
• LocallyFinite.exists_forall_eventually_atTop_eventually_eq' 📋 Mathlib.Topology.LocallyFinite
  {X : Type u_4} [TopologicalSpace X] {π : X → Sort u_6} {f : ℕ → (x : X) → π x} (hf : LocallyFinite fun n => {x | f (n + 1) x ≠ f n x}) : ∃ F, ∀ (x : X), ∀ᶠ (n : ℕ) in Filter.atTop, ∀ᶠ (y : X) in nhds x, f n y = F y
• LocallyFinite.exists_forall_eventually_eq_prod 📋 Mathlib.Topology.LocallyFinite
  {X : Type u_4} [TopologicalSpace X] {π : X → Sort u_6} {f : ℕ → (x : X) → π x} (hf : LocallyFinite fun n => {x | f (n + 1) x ≠ f n x}) : ∃ F, ∀ (x : X), ∀ᶠ (p : ℕ × X) in Filter.atTop ×ˢ nhds x, f p.1 p.2 = F p.2
• LocallyFinite.fintypeOfCompact 📋 Mathlib.Topology.Compactness.LocallyFinite
  {X : Type u_1} {ι : Type u_2} [TopologicalSpace X] [CompactSpace X] {f : ι → Set X} (hf : LocallyFinite f) (hne : ∀ (i : ι), (f i).Nonempty) : Fintype ι
• LocallyFinite.finite_nonempty_of_compact 📋 Mathlib.Topology.Compactness.LocallyFinite
  {X : Type u_1} {ι : Type u_2} [TopologicalSpace X] [CompactSpace X] {f : ι → Set X} (hf : LocallyFinite f) : {i | (f i).Nonempty}.Finite
• LocallyFinite.finite_of_compact 📋 Mathlib.Topology.Compactness.LocallyFinite
  {X : Type u_1} {ι : Type u_2} [TopologicalSpace X] [CompactSpace X] {f : ι → Set X} (hf : LocallyFinite f) (hne : ∀ (i : ι), (f i).Nonempty) : Set.univ.Finite
• LocallyFinite.finite_nonempty_inter_compact 📋 Mathlib.Topology.Compactness.LocallyFinite
  {X : Type u_1} {ι : Type u_2} [TopologicalSpace X] {s : Set X} {f : ι → Set X} (hf : LocallyFinite f) (hs : IsCompact s) : {i | (f i ∩ s).Nonempty}.Finite
• LocallyFinite.encodable 📋 Mathlib.Topology.Compactness.SigmaCompact
  {X : Type u_1} [TopologicalSpace X] [SigmaCompactSpace X] {ι : Type u_4} {f : ι → Set X} (hf : LocallyFinite f) (hne : ∀ (i : ι), (f i).Nonempty) : Encodable ι
• LocallyFinite.countable_univ 📋 Mathlib.Topology.Compactness.SigmaCompact
  {X : Type u_1} {ι : Type u_3} [TopologicalSpace X] [SigmaCompactSpace X] {f : ι → Set X} (hf : LocallyFinite f) (hne : ∀ (i : ι), (f i).Nonempty) : Set.univ.Countable
• locallyFinite_mulSupport_iff 📋 Mathlib.Topology.Algebra.Support
  {X : Type u_1} {M : Type u_7} {ι : Type u_9} [TopologicalSpace X] [One M] {f : ι → X → M} : (LocallyFinite fun i => Function.mulSupport (f i)) ↔ LocallyFinite fun i => mulTSupport (f i)
• locallyFinite_support_iff 📋 Mathlib.Topology.Algebra.Support
  {X : Type u_1} {M : Type u_7} {ι : Type u_9} [TopologicalSpace X] [Zero M] {f : ι → X → M} : (LocallyFinite fun i => Function.support (f i)) ↔ LocallyFinite fun i => tsupport (f i)
• LocallyFinite.smul_right 📋 Mathlib.Topology.Algebra.Support
  {X : Type u_1} {M : Type u_7} {R : Type u_8} {ι : Type u_9} [TopologicalSpace X] [Zero M] [SMulZeroClass R M] {f : ι → X → M} (h : LocallyFinite fun i => Function.support (f i)) (s : ι → X → R) : LocallyFinite fun i => Function.support (s i • f i)
• LocallyFinite.smul_left 📋 Mathlib.Topology.Algebra.Support
  {X : Type u_1} {M : Type u_7} {R : Type u_8} {ι : Type u_9} [TopologicalSpace X] [Zero R] [Zero M] [SMulWithZero R M] {s : ι → X → R} (h : LocallyFinite fun i => Function.support (s i)) (f : ι → X → M) : LocallyFinite fun i => Function.support (s i • f i)
• LocallyFinite.exists_finset_nhds_mulSupport_subset 📋 Mathlib.Topology.Algebra.Support
  {X : Type u_1} {R : Type u_8} {ι : Type u_9} [TopologicalSpace X] {U : ι → Set X} [One R] {f : ι → X → R} (hlf : LocallyFinite fun i => Function.mulSupport (f i)) (hso : ∀ (i : ι), mulTSupport (f i) ⊆ U i) (ho : ∀ (i : ι), IsOpen (U i)) (x : X) : ∃ is, ∃ n ∈ nhds x, n ⊆ ⋂ i ∈ is, U i ∧ ∀ z ∈ n, (Function.mulSupport fun i => f i z) ⊆ ↑is
• LocallyFinite.exists_finset_nhds_support_subset 📋 Mathlib.Topology.Algebra.Support
  {X : Type u_1} {R : Type u_8} {ι : Type u_9} [TopologicalSpace X] {U : ι → Set X} [Zero R] {f : ι → X → R} (hlf : LocallyFinite fun i => Function.support (f i)) (hso : ∀ (i : ι), tsupport (f i) ⊆ U i) (ho : ∀ (i : ι), IsOpen (U i)) (x : X) : ∃ is, ∃ n ∈ nhds x, n ⊆ ⋂ i ∈ is, U i ∧ ∀ z ∈ n, (Function.support fun i => f i z) ⊆ ↑is
• LocallyFinite.exists_finset_mulSupport 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {X : Type u_5} [TopologicalSpace X] {M : Type u_6} [One M] {f : ι → X → M} (hf : LocallyFinite fun i => Function.mulSupport (f i)) (x₀ : X) : ∃ I, ∀ᶠ (x : X) in nhds x₀, (Function.mulSupport fun i => f i x) ⊆ ↑I
• LocallyFinite.exists_finset_support 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {X : Type u_5} [TopologicalSpace X] {M : Type u_6} [Zero M] {f : ι → X → M} (hf : LocallyFinite fun i => Function.support (f i)) (x₀ : X) : ∃ I, ∀ᶠ (x : X) in nhds x₀, (Function.support fun i => f i x) ⊆ ↑I
• finprod_eventually_eq_prod 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {X : Type u_5} [TopologicalSpace X] {M : Type u_6} [CommMonoid M] {f : ι → X → M} (hf : LocallyFinite fun i => Function.mulSupport (f i)) (x : X) : ∃ s, ∀ᶠ (y : X) in nhds x, ∏ᶠ (i : ι), f i y = ∏ i ∈ s, f i y
• finsum_eventually_eq_sum 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {X : Type u_5} [TopologicalSpace X] {M : Type u_6} [AddCommMonoid M] {f : ι → X → M} (hf : LocallyFinite fun i => Function.support (f i)) (x : X) : ∃ s, ∀ᶠ (y : X) in nhds x, ∑ᶠ (i : ι), f i y = ∑ i ∈ s, f i y
• continuous_finprod 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [CommMonoid M] [ContinuousMul M] {f : ι → X → M} (hc : ∀ (i : ι), Continuous (f i)) (hf : LocallyFinite fun i => Function.mulSupport (f i)) : Continuous fun x => ∏ᶠ (i : ι), f i x
• continuous_finsum 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M] {f : ι → X → M} (hc : ∀ (i : ι), Continuous (f i)) (hf : LocallyFinite fun i => Function.support (f i)) : Continuous fun x => ∑ᶠ (i : ι), f i x
• continuous_finprod_cond 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [CommMonoid M] [ContinuousMul M] {f : ι → X → M} {p : ι → Prop} (hc : ∀ (i : ι), p i → Continuous (f i)) (hf : LocallyFinite fun i => Function.mulSupport (f i)) : Continuous fun x => ∏ᶠ (i : ι) (_ : p i), f i x
• continuous_finsum_cond 📋 Mathlib.Topology.Algebra.Monoid
  {ι : Type u_1} {M : Type u_3} {X : Type u_5} [TopologicalSpace X] [TopologicalSpace M] [AddCommMonoid M] [ContinuousAdd M] {f : ι → X → M} {p : ι → Prop} (hc : ∀ (i : ι), p i → Continuous (f i)) (hf : LocallyFinite fun i => Function.support (f i)) : Continuous fun x => ∑ᶠ (i : ι) (_ : p i), f i x
• locallyFinite_Icc_of_tendsto 📋 Mathlib.Topology.Order.AtTopBotIxx
  {X : Type u_1} [LinearOrder X] [TopologicalSpace X] [OrderTopology X] {α : Type u_2} [LinearOrder α] [LocallyFiniteOrder α] [NoMaxOrder X] [NoMinOrder X] {f g : α → X} (hl : Filter.Tendsto f Filter.atTop Filter.atTop) (hu : Filter.Tendsto g Filter.atBot Filter.atBot) : LocallyFinite fun n => Set.Icc (f n) (g n)
• locallyFinite_Ico_of_tendsto 📋 Mathlib.Topology.Order.AtTopBotIxx
  {X : Type u_1} [LinearOrder X] [TopologicalSpace X] [OrderTopology X] {α : Type u_2} [LinearOrder α] [LocallyFiniteOrder α] [NoMaxOrder X] [NoMinOrder X] {l u : α → X} (hl : Filter.Tendsto l Filter.atTop Filter.atTop) (hu : Filter.Tendsto u Filter.atBot Filter.atBot) : LocallyFinite fun n => Set.Ico (l n) (u n)
• locallyFinite_Ioc_of_tendsto 📋 Mathlib.Topology.Order.AtTopBotIxx
  {X : Type u_1} [LinearOrder X] [TopologicalSpace X] [OrderTopology X] {α : Type u_2} [LinearOrder α] [LocallyFiniteOrder α] [NoMaxOrder X] [NoMinOrder X] {l u : α → X} (hl : Filter.Tendsto l Filter.atTop Filter.atTop) (hu : Filter.Tendsto u Filter.atBot Filter.atBot) : LocallyFinite fun n => Set.Ioc (l n) (u n)
• locallyFinite_Ioo_of_tendsto 📋 Mathlib.Topology.Order.AtTopBotIxx
  {X : Type u_1} [LinearOrder X] [TopologicalSpace X] [OrderTopology X] {α : Type u_2} [LinearOrder α] [LocallyFiniteOrder α] [NoMaxOrder X] [NoMinOrder X] {l u : α → X} (hl : Filter.Tendsto l Filter.atTop Filter.atTop) (hu : Filter.Tendsto u Filter.atBot Filter.atBot) : LocallyFinite fun n => Set.Ioo (l n) (u n)
• ContinuousMap.uniformSpace_eq_iInf_precomp_of_cover 📋 Mathlib.Topology.UniformSpace.CompactConvergence
  {α : Type u₁} {β : Type u₂} [TopologicalSpace α] [UniformSpace β] {ι : Type u₃} {δ : ι → Type u_1} [(i : ι) → TopologicalSpace (δ i)] (φ : (i : ι) → C(δ i, α)) (h_proper : ∀ (i : ι), IsProperMap ⇑(φ i)) (h_lf : LocallyFinite fun i => Set.range ⇑(φ i)) (h_cover : ⋃ i, Set.range ⇑(φ i) = Set.univ) : inferInstance = ⨅ i, UniformSpace.comap (fun x => x.comp (φ i)) inferInstance
• LocallyFiniteSupport.locallyFinite_support 📋 Mathlib.Topology.LocallyFinsupp
  {X : Type u_1} [TopologicalSpace X] {Y : Type u_2} [Zero Y] (f : X → Y) (h : LocallyFiniteSupport f) : LocallyFinite fun s => {↑s}
• LocallyFiniteSupport.iff_locallyFinite_support 📋 Mathlib.Topology.LocallyFinsupp
  {X : Type u_1} [TopologicalSpace X] {Y : Type u_2} [Zero Y] (f : X → Y) : (LocallyFinite fun s => {↑s}) ↔ LocallyFiniteSupport f
• contMDiff_finprod 📋 Mathlib.Geometry.Manifold.Algebra.Monoid
  {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {n : WithTop ℕ∞} {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [ContMDiffMul I n G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {f : ι → M → G} (h : ∀ (i : ι), ContMDiff I' I n (f i)) (hfin : LocallyFinite fun i => Function.mulSupport (f i)) : ContMDiff I' I n fun x => ∏ᶠ (i : ι), f i x
• contMDiff_finsum 📋 Mathlib.Geometry.Manifold.Algebra.Monoid
  {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {n : WithTop ℕ∞} {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [ContMDiffAdd I n G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {f : ι → M → G} (h : ∀ (i : ι), ContMDiff I' I n (f i)) (hfin : LocallyFinite fun i => Function.support (f i)) : ContMDiff I' I n fun x => ∑ᶠ (i : ι), f i x
• contMDiffAt_finprod 📋 Mathlib.Geometry.Manifold.Algebra.Monoid
  {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {n : WithTop ℕ∞} {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [ContMDiffMul I n G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {x₀ : M} {f : ι → M → G} (lf : LocallyFinite fun i => Function.mulSupport (f i)) (h : ∀ (i : ι), ContMDiffAt I' I n (f i) x₀) : ContMDiffAt I' I n (fun x => ∏ᶠ (i : ι), f i x) x₀
• contMDiffAt_finsum 📋 Mathlib.Geometry.Manifold.Algebra.Monoid
  {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {n : WithTop ℕ∞} {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [ContMDiffAdd I n G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {x₀ : M} {f : ι → M → G} (lf : LocallyFinite fun i => Function.support (f i)) (h : ∀ (i : ι), ContMDiffAt I' I n (f i) x₀) : ContMDiffAt I' I n (fun x => ∑ᶠ (i : ι), f i x) x₀
• contMDiffOn_finprod 📋 Mathlib.Geometry.Manifold.Algebra.Monoid
  {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {n : WithTop ℕ∞} {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [ContMDiffMul I n G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {f : ι → M → G} (lf : LocallyFinite fun i => Function.mulSupport (f i)) (h : ∀ (i : ι), ContMDiffOn I' I n (f i) s) : ContMDiffOn I' I n (fun x => ∏ᶠ (i : ι), f i x) s
• contMDiffOn_finsum 📋 Mathlib.Geometry.Manifold.Algebra.Monoid
  {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {n : WithTop ℕ∞} {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [ContMDiffAdd I n G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {f : ι → M → G} (lf : LocallyFinite fun i => Function.support (f i)) (h : ∀ (i : ι), ContMDiffOn I' I n (f i) s) : ContMDiffOn I' I n (fun x => ∑ᶠ (i : ι), f i x) s
• contMDiffWithinAt_finprod 📋 Mathlib.Geometry.Manifold.Algebra.Monoid
  {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {n : WithTop ℕ∞} {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [ContMDiffMul I n G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {f : ι → M → G} (lf : LocallyFinite fun i => Function.mulSupport (f i)) {x₀ : M} (h : ∀ (i : ι), ContMDiffWithinAt I' I n (f i) s x₀) : ContMDiffWithinAt I' I n (fun x => ∏ᶠ (i : ι), f i x) s x₀
• contMDiffWithinAt_finsum 📋 Mathlib.Geometry.Manifold.Algebra.Monoid
  {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {n : WithTop ℕ∞} {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [ContMDiffAdd I n G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {s : Set M} {f : ι → M → G} (lf : LocallyFinite fun i => Function.support (f i)) {x₀ : M} (h : ∀ (i : ι), ContMDiffWithinAt I' I n (f i) s x₀) : ContMDiffWithinAt I' I n (fun x => ∑ᶠ (i : ι), f i x) s x₀
• contMDiff_finprod_cond 📋 Mathlib.Geometry.Manifold.Algebra.Monoid
  {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {n : WithTop ℕ∞} {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [CommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [ContMDiffMul I n G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {f : ι → M → G} {p : ι → Prop} (hc : ∀ (i : ι), p i → ContMDiff I' I n (f i)) (hf : LocallyFinite fun i => Function.mulSupport (f i)) : ContMDiff I' I n fun x => ∏ᶠ (i : ι) (_ : p i), f i x
• contMDiff_finsum_cond 📋 Mathlib.Geometry.Manifold.Algebra.Monoid
  {ι : Type u_1} {𝕜 : Type u_2} [NontriviallyNormedField 𝕜] {n : WithTop ℕ∞} {H : Type u_3} [TopologicalSpace H] {E : Type u_4} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {I : ModelWithCorners 𝕜 E H} {G : Type u_5} [AddCommMonoid G] [TopologicalSpace G] [ChartedSpace H G] [ContMDiffAdd I n G] {E' : Type u_6} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {H' : Type u_7} [TopologicalSpace H'] {I' : ModelWithCorners 𝕜 E' H'} {M : Type u_8} [TopologicalSpace M] [ChartedSpace H' M] {f : ι → M → G} {p : ι → Prop} (hc : ∀ (i : ι), p i → ContMDiff I' I n (f i)) (hf : LocallyFinite fun i => Function.support (f i)) : ContMDiff I' I n fun x => ∑ᶠ (i : ι) (_ : p i), f i x
• ContMDiff.finsum_section_of_locallyFinite 📋 Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : M → Type u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] {ι : Type u_7} {t : ι → (x : M) → V x} (ht : LocallyFinite fun i => {x | t i x ≠ 0}) (ht' : ∀ (i : ι), ContMDiff I (I.prod (modelWithCornersSelf 𝕜 F)) n fun x => ⟨x, t i x⟩) : ContMDiff I (I.prod (modelWithCornersSelf 𝕜 F)) n fun x => ⟨x, ∑ᶠ (i : ι), t i x⟩
• ContMDiffAt.finsum_section_of_locallyFinite 📋 Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : M → Type u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] {x₀ : M} {ι : Type u_7} {t : ι → (x : M) → V x} (ht : LocallyFinite fun i => {x | t i x ≠ 0}) (ht' : ∀ (i : ι), ContMDiffAt I (I.prod (modelWithCornersSelf 𝕜 F)) n (fun x => ⟨x, t i x⟩) x₀) : ContMDiffAt I (I.prod (modelWithCornersSelf 𝕜 F)) n (fun x => ⟨x, ∑ᶠ (i : ι), t i x⟩) x₀
• ContMDiff.sum_section_of_locallyFinite 📋 Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : M → Type u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] {ι : Type u_7} {t : ι → (x : M) → V x} (ht : LocallyFinite fun i => {x | t i x ≠ 0}) (ht' : ∀ (i : ι), ContMDiff I (I.prod (modelWithCornersSelf 𝕜 F)) n fun x => ⟨x, t i x⟩) : ContMDiff I (I.prod (modelWithCornersSelf 𝕜 F)) n fun x => ⟨x, ∑' (i : ι), t i x⟩
• ContMDiffOn.finsum_section_of_locallyFinite 📋 Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : M → Type u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] {u : Set M} {ι : Type u_7} {t : ι → (x : M) → V x} (ht : LocallyFinite fun i => {x | t i x ≠ 0}) (ht' : ∀ (i : ι), ContMDiffOn I (I.prod (modelWithCornersSelf 𝕜 F)) n (fun x => ⟨x, t i x⟩) u) : ContMDiffOn I (I.prod (modelWithCornersSelf 𝕜 F)) n (fun x => ⟨x, ∑ᶠ (i : ι), t i x⟩) u
• ContMDiffAt.sum_section_of_locallyFinite 📋 Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : M → Type u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] {x₀ : M} {ι : Type u_7} {t : ι → (x : M) → V x} (ht : LocallyFinite fun i => {x | t i x ≠ 0}) (ht' : ∀ (i : ι), ContMDiffAt I (I.prod (modelWithCornersSelf 𝕜 F)) n (fun x => ⟨x, t i x⟩) x₀) : ContMDiffAt I (I.prod (modelWithCornersSelf 𝕜 F)) n (fun x => ⟨x, ∑' (i : ι), t i x⟩) x₀
• ContMDiffWithinAt.finsum_section_of_locallyFinite 📋 Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : M → Type u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] {u : Set M} {x₀ : M} {ι : Type u_7} {t : ι → (x : M) → V x} (ht : LocallyFinite fun i => {x | t i x ≠ 0}) (ht' : ∀ (i : ι), ContMDiffWithinAt I (I.prod (modelWithCornersSelf 𝕜 F)) n (fun x => ⟨x, t i x⟩) u x₀) : ContMDiffWithinAt I (I.prod (modelWithCornersSelf 𝕜 F)) n (fun x => ⟨x, ∑ᶠ (i : ι), t i x⟩) u x₀
• ContMDiffOn.sum_section_of_locallyFinite 📋 Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : M → Type u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] {u : Set M} {ι : Type u_7} {t : ι → (x : M) → V x} (ht : LocallyFinite fun i => {x | t i x ≠ 0}) (ht' : ∀ (i : ι), ContMDiffOn I (I.prod (modelWithCornersSelf 𝕜 F)) n (fun x => ⟨x, t i x⟩) u) : ContMDiffOn I (I.prod (modelWithCornersSelf 𝕜 F)) n (fun x => ⟨x, ∑' (i : ι), t i x⟩) u
• ContMDiffWithinAt.sum_section_of_locallyFinite 📋 Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection
  {𝕜 : Type u_1} [NontriviallyNormedField 𝕜] {E : Type u_2} [NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type u_3} [TopologicalSpace H] {I : ModelWithCorners 𝕜 E H} {M : Type u_4} [TopologicalSpace M] [ChartedSpace H M] {F : Type u_5} [NormedAddCommGroup F] [NormedSpace 𝕜 F] {n : WithTop ℕ∞} {V : M → Type u_6} [TopologicalSpace (Bundle.TotalSpace F V)] [(x : M) → TopologicalSpace (V x)] [FiberBundle F V] [(x : M) → AddCommGroup (V x)] [(x : M) → Module 𝕜 (V x)] [VectorBundle 𝕜 F V] {u : Set M} {x₀ : M} {ι : Type u_7} {t : ι → (x : M) → V x} (ht : LocallyFinite fun i => {x | t i x ≠ 0}) (ht' : ∀ (i : ι), ContMDiffWithinAt I (I.prod (modelWithCornersSelf 𝕜 F)) n (fun x => ⟨x, t i x⟩) u x₀) : ContMDiffWithinAt I (I.prod (modelWithCornersSelf 𝕜 F)) n (fun x => ⟨x, ∑' (i : ι), t i x⟩) u x₀
• precise_refinement 📋 Mathlib.Topology.Compactness.Paracompact
  {ι : Type u} {X : Type v} [TopologicalSpace X] [ParacompactSpace X] (u : ι → Set X) (uo : ∀ (a : ι), IsOpen (u a)) (uc : ⋃ i, u i = Set.univ) : ∃ v, (∀ (a : ι), IsOpen (v a)) ∧ ⋃ i, v i = Set.univ ∧ LocallyFinite v ∧ ∀ (a : ι), v a ⊆ u a
• ParacompactSpace.locallyFinite_refinement 📋 Mathlib.Topology.Compactness.Paracompact
  {X : Type v} {inst✝ : TopologicalSpace X} [self : ParacompactSpace X] (α : Type v) (s : α → Set X) : (∀ (a : α), IsOpen (s a)) → ⋃ a, s a = Set.univ → ∃ β t, (∀ (b : β), IsOpen (t b)) ∧ ⋃ b, t b = Set.univ ∧ LocallyFinite t ∧ ∀ (b : β), ∃ a, t b ⊆ s a
• ParacompactSpace.mk 📋 Mathlib.Topology.Compactness.Paracompact
  {X : Type v} [TopologicalSpace X] (locallyFinite_refinement : ∀ (α : Type v) (s : α → Set X), (∀ (a : α), IsOpen (s a)) → ⋃ a, s a = Set.univ → ∃ β t, (∀ (b : β), IsOpen (t b)) ∧ ⋃ b, t b = Set.univ ∧ LocallyFinite t ∧ ∀ (b : β), ∃ a, t b ⊆ s a) : ParacompactSpace X
• precise_refinement_set 📋 Mathlib.Topology.Compactness.Paracompact
  {ι : Type u} {X : Type v} [TopologicalSpace X] [ParacompactSpace X] {s : Set X} (hs : IsClosed s) (u : ι → Set X) (uo : ∀ (i : ι), IsOpen (u i)) (us : s ⊆ ⋃ i, u i) : ∃ v, (∀ (i : ι), IsOpen (v i)) ∧ s ⊆ ⋃ i, v i ∧ LocallyFinite v ∧ ∀ (i : ι), v i ⊆ u i
• refinement_of_locallyCompact_sigmaCompact_of_nhds_basis 📋 Mathlib.Topology.Compactness.Paracompact
  {X : Type v} [TopologicalSpace X] [WeaklyLocallyCompactSpace X] [SigmaCompactSpace X] [T2Space X] {ι : X → Type u} {p : (x : X) → ι x → Prop} {B : (x : X) → ι x → Set X} (hB : ∀ (x : X), (nhds x).HasBasis (p x) (B x)) : ∃ α c r, (∀ (a : α), p (c a) (r a)) ∧ ⋃ a, B (c a) (r a) = Set.univ ∧ LocallyFinite fun a => B (c a) (r a)
• ParacompactSpace.of_hasBasis 📋 Mathlib.Topology.Compactness.Paracompact
  {X : Type v} [TopologicalSpace X] {ι : X → Sort u_1} {p : (x : X) → ι x → Prop} {s : (x : X) → ι x → Set X} (hb : ∀ (x : X), (nhds x).HasBasis (p x) (s x)) (h : ∀ (f : (x : X) → ι x), (∀ (x : X), p x (f x)) → ∃ β t, (∀ (b : β), IsOpen (t b)) ∧ ⋃ b, t b = Set.univ ∧ LocallyFinite t ∧ ∀ (b : β), ∃ x, t b ⊆ s x (f x)) : ParacompactSpace X
• refinement_of_locallyCompact_sigmaCompact_of_nhds_basis_set 📋 Mathlib.Topology.Compactness.Paracompact
  {X : Type v} [TopologicalSpace X] [WeaklyLocallyCompactSpace X] [SigmaCompactSpace X] [T2Space X] {ι : X → Type u} {p : (x : X) → ι x → Prop} {B : (x : X) → ι x → Set X} {s : Set X} (hs : IsClosed s) (hB : ∀ x ∈ s, (nhds x).HasBasis (p x) (B x)) : ∃ α c r, (∀ (a : α), c a ∈ s ∧ p (c a) (r a)) ∧ s ⊆ ⋃ a, B (c a) (r a) ∧ LocallyFinite fun a => B (c a) (r a)
• BumpCovering.locallyFinite' 📋 Mathlib.Topology.PartitionOfUnity
  {ι : Type u_1} {X : Type u_2} [TopologicalSpace X] {s : optParam (Set X) Set.univ} (self : BumpCovering ι X s) : LocallyFinite fun i => Function.support ⇑(self.toFun i)
• PartitionOfUnity.locallyFinite' 📋 Mathlib.Topology.PartitionOfUnity
  {ι : Type u_1} {X : Type u_2} [TopologicalSpace X] {s : optParam (Set X) Set.univ} (self : PartitionOfUnity ι X s) : LocallyFinite fun i => Function.support ⇑(self.toFun i)
• BumpCovering.exists_isSubordinate_of_locallyFinite 📋 Mathlib.Topology.PartitionOfUnity
  {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [NormalSpace X] (hs : IsClosed s) (U : ι → Set X) (ho : ∀ (i : ι), IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ i, U i) : ∃ f, f.IsSubordinate U
• PartitionOfUnity.exists_isSubordinate_of_locallyFinite 📋 Mathlib.Topology.PartitionOfUnity
  {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [NormalSpace X] (hs : IsClosed s) (U : ι → Set X) (ho : ∀ (i : ι), IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ i, U i) : ∃ f, f.IsSubordinate U
• BumpCovering.locallyFinite 📋 Mathlib.Topology.PartitionOfUnity
  {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) : LocallyFinite fun i => Function.support ⇑(f i)
• PartitionOfUnity.locallyFinite 📋 Mathlib.Topology.PartitionOfUnity
  {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : PartitionOfUnity ι X s) : LocallyFinite fun i => Function.support ⇑(f i)
• BumpCovering.locallyFinite_tsupport 📋 Mathlib.Topology.PartitionOfUnity
  {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : BumpCovering ι X s) : LocallyFinite fun i => tsupport ⇑(f i)
• PartitionOfUnity.locallyFinite_tsupport 📋 Mathlib.Topology.PartitionOfUnity
  {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} (f : PartitionOfUnity ι X s) : LocallyFinite fun i => tsupport ⇑(f i)
• BumpCovering.exists_isSubordinate_hasCompactSupport_of_locallyFinite_t2space 📋 Mathlib.Topology.PartitionOfUnity
  {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [LocallyCompactSpace X] [T2Space X] (hs : IsCompact s) (U : ι → Set X) (ho : ∀ (i : ι), IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ i, U i) : ∃ f, f.IsSubordinate U ∧ ∀ (i : ι), HasCompactSupport ⇑(f i)
• PartitionOfUnity.exists_isSubordinate_of_locallyFinite_t2space 📋 Mathlib.Topology.PartitionOfUnity
  {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [LocallyCompactSpace X] [T2Space X] (hs : IsCompact s) (U : ι → Set X) (ho : ∀ (i : ι), IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ i, U i) : ∃ f, f.IsSubordinate U ∧ ∀ (i : ι), HasCompactSupport ⇑(f i)
• PartitionOfUnity.mk 📋 Mathlib.Topology.PartitionOfUnity
  {ι : Type u_1} {X : Type u_2} [TopologicalSpace X] {s : optParam (Set X) Set.univ} (toFun : ι → C(X, ℝ)) (locallyFinite' : LocallyFinite fun i => Function.support ⇑(toFun i)) (nonneg' : 0 ≤ toFun) (sum_eq_one' : ∀ x ∈ s, ∑ᶠ (i : ι), (toFun i) x = 1) (sum_le_one' : ∀ (x : X), ∑ᶠ (i : ι), (toFun i) x ≤ 1) : PartitionOfUnity ι X s
• BumpCovering.mk 📋 Mathlib.Topology.PartitionOfUnity
  {ι : Type u_1} {X : Type u_2} [TopologicalSpace X] {s : optParam (Set X) Set.univ} (toFun : ι → C(X, ℝ)) (locallyFinite' : LocallyFinite fun i => Function.support ⇑(toFun i)) (nonneg' : 0 ≤ toFun) (le_one' : toFun ≤ 1) (eventuallyEq_one' : ∀ x ∈ s, ∃ i, ⇑(toFun i) =ᶠ[nhds x] 1) : BumpCovering ι X s
• BumpCovering.exists_isSubordinate_of_locallyFinite_of_prop 📋 Mathlib.Topology.PartitionOfUnity
  {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [NormalSpace X] (p : (X → ℝ) → Prop) (h01 : ∀ (s t : Set X), IsClosed s → IsClosed t → Disjoint s t → ∃ f, p ⇑f ∧ Set.EqOn (⇑f) 0 s ∧ Set.EqOn (⇑f) 1 t ∧ ∀ (x : X), f x ∈ Set.Icc 0 1) (hs : IsClosed s) (U : ι → Set X) (ho : ∀ (i : ι), IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ i, U i) : ∃ f, (∀ (i : ι), p ⇑(f i)) ∧ f.IsSubordinate U
• BumpCovering.exists_isSubordinate_of_locallyFinite_of_prop_t2space 📋 Mathlib.Topology.PartitionOfUnity
  {ι : Type u} {X : Type v} [TopologicalSpace X] {s : Set X} [LocallyCompactSpace X] [T2Space X] (p : (X → ℝ) → Prop) (h01 : ∀ (s t : Set X), IsClosed s → IsCompact t → Disjoint s t → ∃ f, p ⇑f ∧ Set.EqOn (⇑f) 0 s ∧ Set.EqOn (⇑f) 1 t ∧ ∀ (x : X), f x ∈ Set.Icc 0 1) (hs : IsCompact s) (U : ι → Set X) (ho : ∀ (i : ι), IsOpen (U i)) (hf : LocallyFinite U) (hU : s ⊆ ⋃ i, U i) : ∃ f, (∀ (i : ι), p ⇑(f i)) ∧ f.IsSubordinate U ∧ ∀ (i : ι), HasCompactSupport ⇑(f i)
• Metric.eventually_nhds_zero_forall_closedEBall_subset 📋 Mathlib.Topology.MetricSpace.PartitionOfUnity
  {ι : Type u_1} {X : Type u_2} [EMetricSpace X] {K U : ι → Set X} (hK : ∀ (i : ι), IsClosed (K i)) (hU : ∀ (i : ι), IsOpen (U i)) (hKU : ∀ (i : ι), K i ⊆ U i) (hfin : LocallyFinite K) (x : X) : ∀ᶠ (p : ENNReal × X) in nhds 0 ×ˢ nhds x, ∀ (i : ι), p.2 ∈ K i → Metric.closedEBall p.2 p.1 ⊆ U i
• Metric.exists_forall_closedEBall_subset_aux₁ 📋 Mathlib.Topology.MetricSpace.PartitionOfUnity
  {ι : Type u_1} {X : Type u_2} [EMetricSpace X] {K U : ι → Set X} (hK : ∀ (i : ι), IsClosed (K i)) (hU : ∀ (i : ι), IsOpen (U i)) (hKU : ∀ (i : ι), K i ⊆ U i) (hfin : LocallyFinite K) (x : X) : ∃ r, ∀ᶠ (y : X) in nhds x, r ∈ Set.Ioi 0 ∩ ENNReal.ofReal ⁻¹' ⋂ i, ⋂ (_ : y ∈ K i), {r | Metric.closedEBall y r ⊆ U i}
• Metric.exists_continuous_nnreal_forall_closedBall_subset 📋 Mathlib.Topology.MetricSpace.PartitionOfUnity
  {ι : Type u_1} {X : Type u_2} [MetricSpace X] {K U : ι → Set X} (hK : ∀ (i : ι), IsClosed (K i)) (hU : ∀ (i : ι), IsOpen (U i)) (hKU : ∀ (i : ι), K i ⊆ U i) (hfin : LocallyFinite K) : ∃ δ, (∀ (x : X), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, Metric.closedBall x ↑(δ x) ⊆ U i
• Metric.exists_continuous_ennreal_forall_closedEBall_subset 📋 Mathlib.Topology.MetricSpace.PartitionOfUnity
  {ι : Type u_1} {X : Type u_2} [EMetricSpace X] {K U : ι → Set X} (hK : ∀ (i : ι), IsClosed (K i)) (hU : ∀ (i : ι), IsOpen (U i)) (hKU : ∀ (i : ι), K i ⊆ U i) (hfin : LocallyFinite K) : ∃ δ, (∀ (x : X), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, Metric.closedEBall x (δ x) ⊆ U i
• Metric.exists_continuous_nnreal_forall_closedEBall_subset 📋 Mathlib.Topology.MetricSpace.PartitionOfUnity
  {ι : Type u_1} {X : Type u_2} [EMetricSpace X] {K U : ι → Set X} (hK : ∀ (i : ι), IsClosed (K i)) (hU : ∀ (i : ι), IsOpen (U i)) (hKU : ∀ (i : ι), K i ⊆ U i) (hfin : LocallyFinite K) : ∃ δ, (∀ (x : X), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, Metric.closedEBall x ↑(δ x) ⊆ U i
• Metric.exists_continuous_real_forall_closedBall_subset 📋 Mathlib.Topology.MetricSpace.PartitionOfUnity
  {ι : Type u_1} {X : Type u_2} [MetricSpace X] {K U : ι → Set X} (hK : ∀ (i : ι), IsClosed (K i)) (hU : ∀ (i : ι), IsOpen (U i)) (hKU : ∀ (i : ι), K i ⊆ U i) (hfin : LocallyFinite K) : ∃ δ, (∀ (x : X), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, Metric.closedBall x (δ x) ⊆ U i
• Metric.exists_continuous_real_forall_closedEBall_subset 📋 Mathlib.Topology.MetricSpace.PartitionOfUnity
  {ι : Type u_1} {X : Type u_2} [EMetricSpace X] {K U : ι → Set X} (hK : ∀ (i : ι), IsClosed (K i)) (hU : ∀ (i : ι), IsOpen (U i)) (hKU : ∀ (i : ι), K i ⊆ U i) (hfin : LocallyFinite K) : ∃ δ, (∀ (x : X), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, Metric.closedEBall x (ENNReal.ofReal (δ x)) ⊆ U i
• SmoothBumpCovering.locallyFinite 📋 Mathlib.Geometry.Manifold.PartitionOfUnity
  {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] [FiniteDimensional ℝ E] {s : Set M} (fs : SmoothBumpCovering ι I M s) : LocallyFinite fun i => Function.support ↑(fs.toFun i)
• SmoothBumpCovering.locallyFinite' 📋 Mathlib.Geometry.Manifold.PartitionOfUnity
  {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] [FiniteDimensional ℝ E] {s : optParam (Set M) Set.univ} (self : SmoothBumpCovering ι I M s) : LocallyFinite fun i => Function.support ↑(self.toFun i)
• SmoothBumpCovering.mk 📋 Mathlib.Geometry.Manifold.PartitionOfUnity
  {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] [FiniteDimensional ℝ E] {s : optParam (Set M) Set.univ} (c : ι → M) (toFun : (i : ι) → SmoothBumpFunction I (c i)) (c_mem' : ∀ (i : ι), c i ∈ s) (locallyFinite' : LocallyFinite fun i => Function.support ↑(toFun i)) (eventuallyEq_one' : ∀ x ∈ s, ∃ i, ↑(toFun i) =ᶠ[nhds x] 1) : SmoothBumpCovering ι I M s
• SmoothPartitionOfUnity.locallyFinite' 📋 Mathlib.Geometry.Manifold.PartitionOfUnity
  {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : optParam (Set M) Set.univ} (self : SmoothPartitionOfUnity ι I M s) : LocallyFinite fun i => Function.support ⇑(self.toFun i)
• SmoothPartitionOfUnity.locallyFinite 📋 Mathlib.Geometry.Manifold.PartitionOfUnity
  {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : Set M} (f : SmoothPartitionOfUnity ι I M s) : LocallyFinite fun i => Function.support ⇑(f i)
• Metric.exists_contMDiffMap_forall_closedBall_subset 📋 Mathlib.Geometry.Manifold.PartitionOfUnity
  {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] (I : ModelWithCorners ℝ E H) [FiniteDimensional ℝ E] {n : ℕ∞} {M : Type u_1} [MetricSpace M] [ChartedSpace H M] [IsManifold I (↑⊤) M] [SigmaCompactSpace M] {K U : ι → Set M} (hK : ∀ (i : ι), IsClosed (K i)) (hU : ∀ (i : ι), IsOpen (U i)) (hKU : ∀ (i : ι), K i ⊆ U i) (hfin : LocallyFinite K) : ∃ δ, (∀ (x : M), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, Metric.closedBall x (δ x) ⊆ U i
• Metric.exists_contMDiffMap_forall_closedEBall_subset 📋 Mathlib.Geometry.Manifold.PartitionOfUnity
  {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] (I : ModelWithCorners ℝ E H) [FiniteDimensional ℝ E] {n : ℕ∞} {M : Type u_1} [EMetricSpace M] [ChartedSpace H M] [IsManifold I (↑⊤) M] [SigmaCompactSpace M] {K U : ι → Set M} (hK : ∀ (i : ι), IsClosed (K i)) (hU : ∀ (i : ι), IsOpen (U i)) (hKU : ∀ (i : ι), K i ⊆ U i) (hfin : LocallyFinite K) : ∃ δ, (∀ (x : M), 0 < δ x) ∧ ∀ (i : ι), ∀ x ∈ K i, Metric.closedEBall x (ENNReal.ofReal (δ x)) ⊆ U i
• SmoothPartitionOfUnity.mk 📋 Mathlib.Geometry.Manifold.PartitionOfUnity
  {ι : Type uι} {E : Type uE} [NormedAddCommGroup E] [NormedSpace ℝ E] {H : Type uH} [TopologicalSpace H] {I : ModelWithCorners ℝ E H} {M : Type uM} [TopologicalSpace M] [ChartedSpace H M] {s : optParam (Set M) Set.univ} (toFun : ι → ContMDiffMap I (modelWithCornersSelf ℝ ℝ) M ℝ ↑⊤) (locallyFinite' : LocallyFinite fun i => Function.support ⇑(toFun i)) (nonneg' : ∀ (i : ι) (x : M), 0 ≤ (toFun i) x) (sum_eq_one' : ∀ x ∈ s, ∑ᶠ (i : ι), (toFun i) x = 1) (sum_le_one' : ∀ (x : M), ∑ᶠ (i : ι), (toFun i) x ≤ 1) : SmoothPartitionOfUnity ι I M s
• LocallyFinite.Realizer.to_locallyFinite 📋 Mathlib.Data.Analysis.Topology
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] {F : Ctop.Realizer α} {f : β → Set α} (R : LocallyFinite.Realizer F f) : LocallyFinite f
• locallyFinite_iff_exists_realizer 📋 Mathlib.Data.Analysis.Topology
  {α : Type u_1} {β : Type u_2} [TopologicalSpace α] (F : Ctop.Realizer α) {f : β → Set α} : LocallyFinite f ↔ Nonempty (LocallyFinite.Realizer F f)
• MDifferentiable.finsum_section_of_locallyFinite 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 F] [FiberBundle F E] [(x : B) → AddCommGroup (E x)] [(x : B) → Module 𝕜 (E x)] [VectorBundle 𝕜 F E] {HB : Type u_6} [TopologicalSpace HB] [ChartedSpace HB B] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {I : ModelWithCorners 𝕜 EB HB} {ι : Type u_8} {t : ι → (x : B) → E x} (ht : LocallyFinite fun i => {x | t i x ≠ 0}) (ht' : ∀ (i : ι), MDiff fun x => ⟨x, t i x⟩) : MDiff fun x => ⟨x, ∑ᶠ (i : ι), t i x⟩
• MDifferentiableAt.finsum_section_of_locallyFinite 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 F] [FiberBundle F E] [(x : B) → AddCommGroup (E x)] [(x : B) → Module 𝕜 (E x)] [VectorBundle 𝕜 F E] {HB : Type u_6} [TopologicalSpace HB] [ChartedSpace HB B] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {I : ModelWithCorners 𝕜 EB HB} {x₀ : B} {ι : Type u_8} {t : ι → (x : B) → E x} (ht : LocallyFinite fun i => {x | t i x ≠ 0}) (ht' : ∀ (i : ι), (MDiffAt fun x => ⟨x, t i x⟩) x₀) : (MDiffAt fun x => ⟨x, ∑ᶠ (i : ι), t i x⟩) x₀
• MDifferentiable.sum_section_of_locallyFinite 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 F] [FiberBundle F E] [(x : B) → AddCommGroup (E x)] [(x : B) → Module 𝕜 (E x)] [VectorBundle 𝕜 F E] {HB : Type u_6} [TopologicalSpace HB] [ChartedSpace HB B] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {I : ModelWithCorners 𝕜 EB HB} {ι : Type u_8} {t : ι → (x : B) → E x} (ht : LocallyFinite fun i => {x | t i x ≠ 0}) (ht' : ∀ (i : ι), MDiff fun x => ⟨x, t i x⟩) : MDiff fun x => ⟨x, ∑' (i : ι), t i x⟩
• MDifferentiableOn.finsum_section_of_locallyFinite 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 F] [FiberBundle F E] [(x : B) → AddCommGroup (E x)] [(x : B) → Module 𝕜 (E x)] [VectorBundle 𝕜 F E] {HB : Type u_6} [TopologicalSpace HB] [ChartedSpace HB B] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {I : ModelWithCorners 𝕜 EB HB} {u : Set B} {ι : Type u_8} {t : ι → (x : B) → E x} (ht : LocallyFinite fun i => {x | t i x ≠ 0}) (ht' : ∀ (i : ι), MDiff[u] fun x => ⟨x, t i x⟩) : MDiff[u] fun x => ⟨x, ∑ᶠ (i : ι), t i x⟩
• MDifferentiableAt.sum_section_of_locallyFinite 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 F] [FiberBundle F E] [(x : B) → AddCommGroup (E x)] [(x : B) → Module 𝕜 (E x)] [VectorBundle 𝕜 F E] {HB : Type u_6} [TopologicalSpace HB] [ChartedSpace HB B] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {I : ModelWithCorners 𝕜 EB HB} {x₀ : B} {ι : Type u_8} {t : ι → (x : B) → E x} (ht : LocallyFinite fun i => {x | t i x ≠ 0}) (ht' : ∀ (i : ι), (MDiffAt fun x => ⟨x, t i x⟩) x₀) : (MDiffAt fun x => ⟨x, ∑' (i : ι), t i x⟩) x₀
• MDifferentiableWithinAt.finsum_section_of_locallyFinite 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 F] [FiberBundle F E] [(x : B) → AddCommGroup (E x)] [(x : B) → Module 𝕜 (E x)] [VectorBundle 𝕜 F E] {HB : Type u_6} [TopologicalSpace HB] [ChartedSpace HB B] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {I : ModelWithCorners 𝕜 EB HB} {u : Set B} {x₀ : B} {ι : Type u_8} {t : ι → (x : B) → E x} (ht : LocallyFinite fun i => {x | t i x ≠ 0}) (ht' : ∀ (i : ι), (MDiffAt[u] fun x => ⟨x, t i x⟩) x₀) : (MDiffAt[u] fun x => ⟨x, ∑ᶠ (i : ι), t i x⟩) x₀
• MDifferentiableOn.sum_section_of_locallyFinite 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 F] [FiberBundle F E] [(x : B) → AddCommGroup (E x)] [(x : B) → Module 𝕜 (E x)] [VectorBundle 𝕜 F E] {HB : Type u_6} [TopologicalSpace HB] [ChartedSpace HB B] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {I : ModelWithCorners 𝕜 EB HB} {u : Set B} {ι : Type u_8} {t : ι → (x : B) → E x} (ht : LocallyFinite fun i => {x | t i x ≠ 0}) (ht' : ∀ (i : ι), MDiff[u] fun x => ⟨x, t i x⟩) : MDiff[u] fun x => ⟨x, ∑' (i : ι), t i x⟩
• MDifferentiableWithinAt.sum_section_of_locallyFinite 📋 Mathlib.Geometry.Manifold.VectorBundle.MDifferentiable
  {𝕜 : Type u_1} {B : Type u_2} {F : Type u_3} {E : B → Type u_5} [TopologicalSpace B] [TopologicalSpace (Bundle.TotalSpace F E)] [(x : B) → TopologicalSpace (E x)] [NormedAddCommGroup F] [NontriviallyNormedField 𝕜] [NormedSpace 𝕜 F] [FiberBundle F E] [(x : B) → AddCommGroup (E x)] [(x : B) → Module 𝕜 (E x)] [VectorBundle 𝕜 F E] {HB : Type u_6} [TopologicalSpace HB] [ChartedSpace HB B] {EB : Type u_7} [NormedAddCommGroup EB] [NormedSpace 𝕜 EB] {I : ModelWithCorners 𝕜 EB HB} {u : Set B} {x₀ : B} {ι : Type u_8} {t : ι → (x : B) → E x} (ht : LocallyFinite fun i => {x | t i x ≠ 0}) (ht' : ∀ (i : ι), (MDiffAt[u] fun x => ⟨x, t i x⟩) x₀) : (MDiffAt[u] fun x => ⟨x, ∑' (i : ι), t i x⟩) x₀
• exists_locallyFinite_iUnion_eq_ball_radius_lt 📋 Mathlib.Topology.MetricSpace.ShrinkingLemma
  {α : Type u} [MetricSpace α] [ProperSpace α] {R : α → ℝ} (hR : ∀ (x : α), 0 < R x) : ∃ ι c r r', (∀ (i : ι), 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧ (LocallyFinite fun i => Metric.ball (c i) (r' i)) ∧ ⋃ i, Metric.ball (c i) (r i) = Set.univ
• exists_locallyFinite_subset_iUnion_ball_radius_lt 📋 Mathlib.Topology.MetricSpace.ShrinkingLemma
  {α : Type u} [MetricSpace α] [ProperSpace α] {s : Set α} (hs : IsClosed s) {R : α → ℝ} (hR : ∀ x ∈ s, 0 < R x) : ∃ ι c r r', (∀ (i : ι), c i ∈ s ∧ 0 < r i ∧ r i < r' i ∧ r' i < R (c i)) ∧ (LocallyFinite fun i => Metric.ball (c i) (r' i)) ∧ s ⊆ ⋃ i, Metric.ball (c i) (r i)

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

1. By constant:
   🔍 Real.sin
   finds all lemmas whose statement somehow mentions the sine function.

2. By lemma name substring:
   🔍 "differ"
   finds all lemmas that have "differ" somewhere in their lemma name.

3. 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

4. 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 find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

5. You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

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, _ * _, _ ^ _, |- _ < _ → _
would 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 provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision ce5dd8c