Documentation Verification Report

Induced

📁 Source: Mathlib/Topology/Defs/Induced.lean

Statistics

MetricCount
Definitionscoinduced, induced, IsClosedEmbedding, IsCoherentWith, IsEmbedding, IsInducing, IsOpenEmbedding, IsQuotientMap, instTopologicalSpaceSubtype
9
TheoremsisClosed_range, toIsEmbedding, isOpen_of_forall_induced, injective, toIsInducing, eq_induced, isOpen_range, toIsEmbedding, eq_coinduced, surjective, isClosedEmbedding_iff, isEmbedding_iff, isInducing_iff, isOpenEmbedding_iff, isQuotientMap_iff'
15
Total24

TopologicalSpace

Definitions

NameCategoryTheorems
coinduced 📖CompOp
38 mathmath: coinduced_bot, continuous_coinduced_dom, MeasureTheory.LevyProkhorov.eq_convergenceInDistribution, deltaGenerated_eq_coinduced, TopCat.colimit_topology, Topology.IsGeneratedBy.coinduced, LocPathConnectedSpace.coinduced, Topology.isQuotientMap_iff', coinduced_nhdsAdjoint, Topology.IsQuotientMap.eq_coinduced, coinduced_sSup, coinduced_eq_induced_of_isOpenQuotientMap_of_isInducing, alexandrovDiscrete_coinduced, coinduced_le_iff_le_induced, Equiv.induced_symm, TopCat.coinduced_of_isColimit, ModuleTopology.eq_coinduced_of_surjectiveₛₗ, ModuleTopology.eq_coinduced_of_surjective, DeltaGeneratedSpace.coinduced, coinduced_iSup, isOpen_coinduced, coinduced_id, Equiv.coinduced_symm, coinduced_compose, coinduced_sup, TopCat.nonempty_isColimit_iff_eq_coinduced, MeasureTheory.LevyProkhorov.le_convergenceInDistribution, continuous_iff_coinduced_le, MeasureTheory.levyProkhorov_le_convergenceInDistribution, RestrictedProduct.topologicalSpace_eq_iSup, coinduced_mono, SequentialSpace.coinduced, continuous_coinduced_rng, gc_coinduced_induced, isClosed_coinduced, Homeomorph.coinduced_eq, Continuous.coinduced_le, MeasureTheory.levyProkhorov_eq_convergenceInDistribution
induced 📖CompOp
98 mathmath: ContinuousMap.compactOpen_eq_iInf_induced, isOpen_induced_eq, induced_inf, eq_induced_by_maps_to_sierpinski, Homeomorph.induced_eq, map_nhds_induced_of_surjective, map_nhdsSet_induced_eq, induced_const, prod_induced_induced, induced_id, TopCat.pullback_topology, induced_iff_nhds_eq, UniformSpace.toTopologicalSpace_comap, Pi.induced_restrict, induced_topology_pure, TopCat.prod_topology, IsTopologicalBasis.inf_induced, Topology.IsEmbedding.induced, induced_orderTopology', continuousMul_induced, induced_sInf, TopCat.nonempty_isLimit_iff_eq_induced, pullbackTopology_def, induced_compose, le_induced_generateFrom, isClosed_induced_iff', PolynormableSpace.induced, UniformConvergenceCLM.topologicalSpace_eq, LinearMap.mem_span_iff_continuous_of_finite, coinduced_eq_induced_of_isOpenQuotientMap_of_isInducing, R1Space.induced, StrictMono.induced_topology_eq_preorder, UniformOnFun.topologicalSpace_eq, ContinuousInv.induced, induced_topology_eq_preorder, continuous_iff_le_induced, coinduced_le_iff_le_induced, RestrictedProduct.topologicalSpace_eq_of_principal, Equiv.induced_symm, continuousSMul_induced, isClosed_induced_iff, topologicalAddGroup_induced, continuousAdd_induced, isOpen_induced, Pi.induced_precomp, secondCountableTopology_induced, completelyRegularSpace_induced, mem_nhds_induced, topologicalGroup_induced, Units.topology_eq_inf, induced_topology_le_preorder, induced_generateFrom_eq, Equiv.polishSpace_induced, Pi.induced_restrict_sUnion, Equiv.coinduced_symm, Continuous.le_induced, nhds_induced, regularSpace_induced, induced_to_pi, continuous_induced_dom, TopCat.induced_of_isLimit, IsTopologicalBasis.induced, continuousSMul_inducedₛₗ, Topology.isInducing_iff, LocallyConvexSpace.induced, inducing_iInf_to_pi, mem_nhdsSet_induced, RestrictedProduct.topologicalSpace_eq_of_bot, nhdsSet_induced, induced_mono, Pi.induced_precomp', closure_induced, map_nhds_induced_eq, Function.Injective.isEmbedding_induced, UniformOnFun.continuousSMul_submodule_of_image_bounded, ContinuousMap.compactOpen_le_induced, induced_top, RestrictedProduct.topologicalSpace_eq_of_top, Topology.IsInducing.eq_induced, borel_comap, map_nhds_induced_of_mem, induced_iInf, isClosed_induced, ContinuousNeg.induced, TopCat.limit_topology, ContinuousMul.induced, isOpen_induced_iff, LinearMap.mem_span_iff_continuous, AddUnits.topology_eq_inf, ContinuousAdd.induced, firstCountableTopology_induced, gc_coinduced_induced, Topology.IsInducing.induced, IsTopologicalBasis.iInf_induced, ContinuousSMul.induced, LinearMap.withSeminorms_induced, continuous_induced_rng, induced_orderTopology

Topology

Definitions

NameCategoryTheorems
IsClosedEmbedding 📖CompData
81 mathmath: CommRingCat.HomTopology.isClosedEmbedding_precomp_of_surjective, Matrix.IsHermitian.isClosedEmbedding_cfcAux, ContinuousAlternatingMap.isClosedEmbedding_toContinuousMultilinearMap, IsClosedEmbedding.inclusion, PrimeSpectrum.isClosedEmbedding_comap_snd, AlgebraicGeometry.isClosedEmbedding_isZariskiLocalAtTarget, isClosedEmbedding_cfcₙAux, IsClosedEmbedding.isClosedEmbedding_iff_continuous_injective_isClosedMap, NonUnitalStarAlgebra.elemental.isClosedEmbedding_coe, NonUnitalAlgebra.elemental.isClosedEmbedding_coe, AlgebraicGeometry.IsClosedImmersion.base_closed, Algebra.elemental.isClosedEmbedding_coe, cfcHom_isClosedEmbedding, Matrix.SpecialLinearGroup.isClosedEmbedding_toGL, EReal.isClosedEmbedding_coe_ennreal, Continuous.isClosedEmbedding, isClosedEmbedding_of_spaced_out, IsClosedEmbedding.id, UniformSpace.hausdorff.isClosedEmbedding_singleton, LinearMap.isClosedEmbedding_of_injective, Dilation.isClosedEmbedding, isSeparatedMap_iff_isClosedEmbedding, TopologicalSpace.IsOpenCover.isClosedEmbedding_iff_restrictPreimage, IsClosedEmbedding.sigmaMk, FiberBundle.totalSpaceMk_isClosedEmbedding, AlgebraicGeometry.instRespectsIsoSchemeTopologicallyIsClosedEmbedding, StarSubalgebra.isClosedEmbedding_inclusion, PrimeSpectrum.isClosedEmbedding_comap_fst, DomMulAct.isClosedEmbedding_mk, Int.isClosedEmbedding_coe_real, ContinuousFunctionalCalculus.exists_cfc_of_predicate, Matrix.SpecialLinearGroup.isClosedEmbedding_mapGLInt, IsHomeomorph.isClosedEmbedding, isClosedEmbedding_update, CommRingCat.HomTopology.isClosedEmbedding_hom, AntilipschitzWith.isClosedEmbedding, DomMulAct.isClosedEmbedding_mk_symm, ContinuousAddMonoidHom.isClosedEmbedding_toContinuousMap, IsClosedEmbedding.subtypeVal, cfcₙHom_isClosedEmbedding, Nat.isClosedEmbedding_coe_rat, IsClosedEmbedding.inr, isClosedEmbedding_iff, IsClosedEmbedding.of_isEmbedding_isClosedMap, AlgebraicGeometry.IsClosedImmersion.eq_inf, IsClosedEmbedding.inl, IsClosedEmbedding.of_continuous_injective_isClosedMap, AbsoluteValue.Completion.isClosedEmbedding_extensionEmbedding_of_comp, isClosedEmbedding_cfcₙHom_of_cfcHom, AlgebraicGeometry.IsClosedImmersion.isClosedEmbedding, NonUnitalContinuousFunctionalCalculus.exists_cfc_of_predicate, StarAlgebra.elemental.isClosedEmbedding_coe, Isometry.isClosedEmbedding, AlgebraicGeometry.Scheme.Hom.isClosedEmbedding, Units.isClosedEmbedding_embedProduct, IsUniformEmbedding.isClosedEmbedding, ContinuousAddMonoidHom.isClosedEmbedding_coe, ContinuousMonoidHom.isClosedEmbedding_toContinuousMap, Nat.isClosedEmbedding_coe_real, AlgebraicGeometry.isClosedImmersion_iff, exists_embedding_euclidean_of_compact, IsClosed.isClosedEmbedding_subtypeVal, Metric.isClosedEmbedding_of_pairwise_le_dist, ModelWithCorners.isClosedEmbedding, TopologicalSpace.Closeds.isClosedEmbedding_singleton, Function.LeftInverse.isClosedEmbedding, Profinite.Nobeling.isClosedEmbedding, Homeomorph.isClosedEmbedding, DomAddAct.isClosedEmbedding_mk, TopologicalSpace.NonemptyCompacts.isClosedEmbedding_toCompacts, isClosedEmbedding_smul_left, PrimeSpectrum.isClosedEmbedding_comap_of_surjective, AddUnits.isClosedEmbedding_embedProduct, DomAddAct.isClosedEmbedding_mk_symm, IsClosedEmbedding.uliftDown, LightProfinite.isClosedEmbedding_natUnionInftyEmbedding, ContinuousMapZero.isClosedEmbedding_toContinuousMap, Int.isClosedEmbedding_coe_rat, ContinuousMonoidHom.isClosedEmbedding_coe, CategoryTheory.PreGaloisCategory.autEmbedding_isClosedEmbedding, Complex.closedEmbedding_intCast
IsCoherentWith 📖CompData
5 mathmath: IsCoherentWith.of_nhds, IsCoherentWith.of_continuous_prop, CompactlyCoherentSpace.isCoherentWith, IsCoherentWith.of_isClosed, IsCoherentWith.of_seq
IsEmbedding 📖CompData
130 mathmath: UniformConvergenceCLM.isEmbedding_coeFn, AbsoluteValue.IsEquiv.isEmbedding_equivWithAbs, CommRingCat.HomTopology.isEmbedding_precomp_of_surjective, TrivSqZeroExt.IsEmbedding.inr, ContinuousMapZero.isEmbedding_toContinuousMap, ContinuousAlternatingMap.isEmbedding_restrictScalars, WeakBilin.isEmbedding, IsHomeomorph.isEmbedding, TopologicalSpace.Compacts.isEmbedding_coe, IsOpenEmbedding.toIsEmbedding, IsEmbedding.inl, IsEmbedding.inclusion, LinearIsometry.isEmbedding, Manifold.IsSmoothEmbedding.isEmbedding, Homeomorph.isEmbedding, AlgebraicGeometry.IsPreimmersion.isEmbedding, BoundedContinuousFunction.isEmbedding_coeFn, UniformOnFun.isEmbedding_toFun_finite, TopologicalSpace.NonemptyCompacts.isEmbedding_coe, isEmbedding_sumElim, IsEmbedding.mk', ContinuousMultilinearMap.isEmbedding_restrictScalars, isEmbedding_iff, PrimeSpectrum.isEmbedding_tensorProductTo_of_surjectiveOnStalks, ContinuousAlternatingMap.isEmbedding_toContinuousMultilinearMap, AntilipschitzWith.isEmbedding, IsOpenEmbedding.isEmbedding, Units.isEmbedding_val_mk', isEmbedding_prodMkLeft, NNReal.isEmbedding_coe, isEmbedding_graph, Complex.UnitDisc.isEmbedding_coe, AlgebraicGeometry.IsPreimmersion.Spec_map_iff, IsEmbedding.induced, isHomeomorph_iff_isEmbedding_surjective, t35Space_iff_isEmbedding_stoneCechUnit, isEmbedding_sigmaMap, PointwiseConvergenceCLM.isEmbedding_coeFn, isEmbedding_stoneCechUnit, TopologicalSpace.productOfMemOpens_isEmbedding, ContinuousMap.isEmbedding_sigmaMk_comp, CommRingCat.HomTopology.isEmbedding_pushout, TopologicalSpace.NonemptyCompacts.isEmbedding_toCloseds, Function.Surjective.isEmbedding_comp, IsEmbedding.sigmaMk, isOpenEmbedding_iff_isEmbedding_isOpenMap, RestrictedProduct.isEmbedding_coe_of_principal, AlgebraicGeometry.SurjectiveOnStalks.isEmbedding_pullback, IsEmbedding.toPullbackDiag, AddUnits.isEmbedding_val_mk', ContinuousLinearMapWOT.isEmbedding_inducingFn, Units.isEmbedding_embedProduct, DomMulAct.isEmbedding_mk, CommRingCat.HomTopology.isEmbedding_hom, IsEmbedding.uliftDown, IsClosedEmbedding.isEmbedding, MeasureTheory.FiniteMeasure.Topology.IsClosedEmbedding.isEmbedding_map_finiteMeasure, Isometry.isEmbedding, Metric.PiNatEmbed.exists_closed_embedding_to_hilbert_cube, AlgebraicGeometry.Scheme.Hom.isEmbedding, IsInducing.isEmbedding, IsEmbedding.id, FiberBundle.totalSpaceMk_isEmbedding, IsClosedEmbedding.toIsEmbedding, isClosedEmbedding_iff, Matrix.SpecialLinearGroup.isEmbedding_toGL, RestrictedProduct.isEmbedding_inclusion_top, ContinuousMultilinearMap.isEmbedding_toUniformOnFun, MeasureTheory.isEmbedding_diracProba, AddUnits.isEmbedding_val, Units.embedding_val_mk, WithTop.isEmbedding_coe, Units.isEmbedding_val, AlgebraicGeometry.isPreimmersion_eq_inf, AlgebraicGeometry.isPreimmersion_iff, ContinuousLinearMap.isEmbedding_restrictScalars, ContinuousMonoidHom.isEmbedding_toContinuousMap, StrictMono.isEmbedding_of_ordConnected, AlgebraicGeometry.IsPreimmersion.SpecMap_iff, TopCat.isEmbedding_pullback_to_prod, DomAddAct.isEmbedding_mk, AlgebraicGeometry.IsPreimmersion.base_embedding, exists_topology_isEmbedding_nat, EReal.isEmbedding_coe_ennreal, ContinuousAddMonoidHom.isEmbedding_toContinuousMap, TrivSqZeroExt.IsEmbedding.inl, isEmbedding_iff_isInducing, Units.isEmbedding_val₀, ENat.isEmbedding_natCast, RestrictedProduct.isEmbedding_coe_of_top, TopologicalSpace.IsOpenCover.isEmbedding_iff_restrictPreimage, AddUnits.embedding_val_mk, SeparationQuotient.isEmbedding_outCLM, TopologicalSpace.Compacts.isEmbedding_toCloseds, PrimeSpectrum.isEmbedding_comap_of_surjective, TopologicalSpace.exists_embedding_l_infty, IsDenseEmbedding.isEmbedding, IsUniformEmbedding.isEmbedding, IsEmbedding.of_leftInverse, MeasureTheory.FiniteMeasure.isEmbedding_toWeakDualBCNN, MeasureTheory.ProbabilityMeasure.toFiniteMeasure_isEmbedding, RestrictedProduct.isEmbedding_inclusion_principal, OrderEmbedding.isEmbedding_of_ordConnected, RestrictedProduct.isEmbedding_coe_of_bot, EReal.isEmbedding_coe, DomMulAct.isEmbedding_mk_symm, TopologicalSpace.NonemptyCompacts.isEmbedding_toCompacts, Function.Injective.isEmbedding_induced, IsEmbedding.subtypeVal, isEmbedding_sigmoid, Rat.isEmbedding_coe_real, Dilation.isEmbedding, Manifold.isSmoothEmbedding_iff, PrimeSpectrum.localization_comap_isEmbedding, isOpenEmbedding_iff, DomAddAct.isEmbedding_mk_symm, Function.LeftInverse.isEmbedding, StarSubalgebra.isEmbedding_inclusion, ENNReal.isEmbedding_coe, TopologicalSpace.Closeds.isEmbedding_singleton, UpperHalfPlane.isEmbedding_coe, AlgebraicGeometry.isEmbedding_isZariskiLocalAtTarget, AddUnits.isEmbedding_embedProduct, AlgebraicGeometry.instRespectsIsoSchemeTopologicallyIsEmbedding, IsEmbedding.of_subsingleton, isEmbedding_of_isOpenQuotientMap_of_isInducing, isEmbedding_prodMkRight, RestrictedProduct.isEmbedding_structureMap, Metric.PiNatEmbed.exists_embedding_to_hilbert_cube, IsEmbedding.inr
IsInducing 📖CompData
61 mathmath: TopCat.isInducing_pullback_to_prod, TopologicalSpace.IsOpenCover.isInducing_iff_restrictPreimage, Homeomorph.isInducing, IsClosedEmbedding.isInducing, IsInducing.id, Equicontinuous.inducing_uniformFun_iff_pi, IsTopologicalAddGroup.isInducing_iff_nhds_zero, PrimeSpectrum.localization_comap_isInducing, ContinuousMonoidHom.isInducing_toContinuousMap, IsInducing.of_subsingleton, Filter.isInducing_nhds, SeparationQuotient.isInducing_mk, isEmbedding_iff, EquicontinuousOn.inducing_uniformOnFun_iff_pi', isInducing_sigmaMap, DomMulAct.isInducing_mk_symm, inducing_sigma, DomAddAct.isInducing_mk, Bundle.Trivial.isInducing_toProd, AntilipschitzWith.isInducing, isInducing_prodMkLeft, isInducing_stoneCechUnit, FiberBundle.Prod.isInducing_diag, FiberPrebundle.totalSpaceMk_isInducing, isInducing_sumElim, PointwiseConvergenceCLM.isInducing_inducingFn, IsUniformInducing.isInducing, ContinuousLinearMapWOT.isInducing_inducingFn, TopologicalSpace.productOfMemOpens_isInducing, IsDenseInducing.isInducing, IsEmbedding.isInducing, IsDenseInducing.toIsInducing, ContinuousAddMonoidHom.isInducing_toContinuousMap, VectorPrebundle.totalSpaceMk_isInducing, isInducing_iff_nhds, TopologicalSpace.exists_isInducing_l_infty, IsOpenEmbedding.isInducing, inducing_pullbackTotalSpaceEmbedding, isInducing_prodMkRight, DomMulAct.isInducing_mk, FiberBundle.totalSpaceMk_isInducing', isEmbedding_iff_isInducing, isInducing_iff, isInducing_const_prod, inducing_iInf_to_pi, completelyRegularSpace_iff_isInducing_stoneCechUnit, IsEmbedding.toIsInducing, EquicontinuousOn.isInducing_uniformOnFun_iff_pi, FiberBundle.totalSpaceMk_isInducing, isInducing_prod_const, IsHomeomorph.isInducing, IsInducing.sumSwap, Matrix.SpecialLinearGroup.isInducing_toGL, Units.isInducing_embedProduct, IsInducing.subtypeVal, AddUnits.isInducing_embedProduct, PrimeSpectrum.comap_isInducing_of_surjective, IsInducing.induced, DomAddAct.isInducing_mk_symm, BoundedContinuousFunction.isInducing_coeFn, IsTopologicalGroup.isInducing_iff_nhds_one
IsOpenEmbedding 📖CompData
73 mathmath: TopCat.binaryCofan_isColimit_iff, IsLocalHomeomorph.isOpenEmbedding_of_injective, TopCat.isOpenEmbedding_iff_comp_isIso, IsOpen.isOpenEmbedding_subtypeVal, UpperHalfPlane.isOpenEmbedding_coe, ENNReal.isOpenEmbedding_coe, DomAddAct.isOpenEmbedding_mk, TopCat.isOpenEmbedding_iff, PrimeSpectrum.localization_away_isOpenEmbedding, IsOpenEmbedding.sumSwap, AlgebraicGeometry.IsOpenImmersion.iff_isIso_stalkMap, EReal.isOpenEmbedding_coe, RestrictedProduct.isOpenEmbedding_structureMap, DomAddAct.isOpenEmbedding_mk_symm, Real.isOpenEmbedding_exp, TopCat.isOpenEmbedding_iff_isIso_comp', AlgebraicGeometry.isOpenEmbedding_isZariskiLocalAtTarget, TopologicalSpace.Opens.isOpenEmbedding', isOpenEmbedding_stereographic_symm, AlgebraicGeometry.Scheme.Cover.isOpenEmbedding_fromGlued, isLocalHomeomorph_iff_isOpenEmbedding_restrict, AlgebraicGeometry.instRespectsIsoSchemeTopologicallyIsOpenEmbedding, CompHausLike.finiteCoproduct.isOpenEmbedding_ι, WithTop.isOpenEmbedding_coe, CompHausLike.Sigma.isOpenEmbedding_ι, IsOpenEmbedding.of_continuous_injective_isOpenMap, IsOpenUnits.isOpenEmbedding_unitsVal, isOpenEmbedding_iff_isEmbedding_isOpenMap, TopCat.GlueData.f_open, OpenPartialHomeomorph.isOpenEmbedding_restrict, TopCat.isOpenEmbedding_iff_comp_isIso', TopCat.isOpenEmbedding_f_zeroHypercover, TopologicalSpace.Opens.isOpenEmbedding, AlgebraicGeometry.isOpenImmersion_iff_stalk, isLocalHomeomorphOn_iff_isOpenEmbedding_restrict, TopCat.GlueData.fromOpenSubsetsGlue_isOpenEmbedding, IsOpenEmbedding.of_isEmbedding, IsOpenEmbedding.inr, AlgebraicGeometry.Scheme.Hom.isOpenEmbedding, AlgebraicGeometry.PresheafedSpace.GlueData.ι_isOpenEmbedding, OnePoint.isOpenEmbedding_coe, TopologicalSpace.NonemptyCompacts.isOpenEmbedding_toCompacts, AlgebraicGeometry.isOpenImmersion_eq_inf, TopCat.GlueData.ι_isOpenEmbedding, isOpenEmbedding_iff_continuous_injective_isOpenMap, isOpenUnits_iff, IsOpenEmbedding.of_isEmbedding_isOpenMap, TopCat.isOpenEmbedding_iff_isIso_comp, IsHomeomorph.isOpenEmbedding, TopologicalSpace.OpenNhds.isOpenEmbedding, IsOpenEmbedding.of_isEmpty, IsLocallyInjective_iff_isOpenEmbedding, IsOpenEmbedding.inl, AlgebraicGeometry.IsOpenImmersion.iff_stalk_iso, AlgebraicGeometry.Scheme.Cover.fromGlued_isOpenEmbedding, IsOpenEmbedding.sigmaMk, Homeomorph.isOpenEmbedding, DomMulAct.isOpenEmbedding_mk_symm, AlgebraicGeometry.SheafedSpace.IsOpenImmersion.sigma_ι_isOpenEmbedding, IsOpenEmbedding.id, IsOpenEmbedding.inclusion, TopologicalSpace.Opens.isOpenEmbedding_of_le, PadicInt.isOpenEmbedding_coe, DomMulAct.isOpenEmbedding_mk, AlgebraicGeometry.PresheafedSpace.IsOpenImmersion.base_open, OpenPartialHomeomorph.to_isOpenEmbedding, isOpenEmbedding_sigmaMap, TopologicalSpace.IsOpenCover.isOpenEmbedding_iff_restrictPreimage, Units.isOpenEmbedding_val, isOpenEmbedding_iff, IsEmbedding.isOpenEmbedding_of_surjective, ENat.isOpenEmbedding_natCast, RestrictedProduct.isOpenEmbedding_inclusion_principal
IsQuotientMap 📖CompData
47 mathmath: IsHomeomorph.isQuotientMap, IsOpenMap.isQuotientMap, QuotientGroup.isQuotientMap_mk, SeparationQuotient.isQuotientMap_prodMap_mk, isQuotientCoveringMap_iff, isQuotientMap_snd, isQuotientMap_quotient_mk', IsQuotientMap.id, isQuotientMap_projIcc, isQuotientMap_iff', isQuotientMap_iff, SeparationQuotient.isQuotientMap_mk, isQuotientMap_iff_isClosed, QuotientRing.isQuotientMap_coe_coe, DomMulAct.isQuotientMap_mk_symm, isQuotientMap_of_isOpenQuotientMap_of_isInducing, FiberBundle.isQuotientMap_proj, IsInducing.isQuotientMap_of_surjective, PrimeSpectrum.isQuotientMap_of_generalizingMap, IsQuotientMap.of_surjective_continuous, IsModuleTopology.isQuotientMap_of_surjective, IsCoveringMap.isQuotientMap, DomAddAct.isQuotientMap_mk_symm, isQuotientMap_fst, QuotientAddGroup.isQuotientMap_mk, IsClosedMap.isQuotientMap, DiscreteQuotient.proj_isQuotientMap, Complex.isQuotientMap_im, PrimeSpectrum.isQuotientMap_of_specializingMap, ConnectedComponents.isQuotientMap_coe, IsQuotientCoveringMap.toIsQuotientMap, Complex.isQuotientMap_re, IsQuotientMap.of_inverse, TopCat.isQuotientMap_of_isColimit_cofork, IsHomeomorphicTrivialFiberBundle.isQuotientMap_proj, AlgebraicGeometry.Flat.isQuotientMap_of_surjective, TopCat.effectiveEpi_iff_isQuotientMap, DomMulAct.isQuotientMap_mk, IsModuleTopology.isQuotientMap_of_surjectiveₛₗ, IsOpenQuotientMap.iff_isOpenMap_isQuotientMap, IsOpenQuotientMap.isQuotientMap, isAddQuotientCoveringMap_iff, ContinuousLinearMap.isQuotientMap, DomAddAct.isQuotientMap_mk, IsAddQuotientCoveringMap.toIsQuotientMap, Homeomorph.isQuotientMap, isQuotientMap_quot_mk

Theorems

NameKindAssumesProvesValidatesDepends On
isClosedEmbedding_iff 📖mathematicalIsClosedEmbedding
IsEmbedding
IsClosed
Set.range
isEmbedding_iff 📖mathematicalIsEmbedding
IsInducing
isInducing_iff 📖mathematicalIsInducing
TopologicalSpace.induced
isOpenEmbedding_iff 📖mathematicalIsOpenEmbedding
IsEmbedding
IsOpen
Set.range
isQuotientMap_iff' 📖mathematicalIsQuotientMap
TopologicalSpace.coinduced

Topology.IsClosedEmbedding

Theorems

NameKindAssumesProvesValidatesDepends On
isClosed_range 📖mathematicalTopology.IsClosedEmbeddingIsClosed
Set.range
toIsEmbedding 📖mathematicalTopology.IsClosedEmbeddingTopology.IsEmbedding

Topology.IsCoherentWith

Theorems

NameKindAssumesProvesValidatesDepends On
isOpen_of_forall_induced 📖Topology.IsCoherentWith
IsOpen
Set.Elem
instTopologicalSpaceSubtype
Set
Set.instMembership
Set.preimage

Topology.IsEmbedding

Theorems

NameKindAssumesProvesValidatesDepends On
injective 📖Topology.IsEmbedding
toIsInducing 📖mathematicalTopology.IsEmbeddingTopology.IsInducing

Topology.IsInducing

Theorems

NameKindAssumesProvesValidatesDepends On
eq_induced 📖mathematicalTopology.IsInducingTopologicalSpace.induced

Topology.IsOpenEmbedding

Theorems

NameKindAssumesProvesValidatesDepends On
isOpen_range 📖mathematicalTopology.IsOpenEmbeddingIsOpen
Set.range
toIsEmbedding 📖mathematicalTopology.IsOpenEmbeddingTopology.IsEmbedding

Topology.IsQuotientMap

Theorems

NameKindAssumesProvesValidatesDepends On
eq_coinduced 📖mathematicalTopology.IsQuotientMapTopologicalSpace.coinduced
surjective 📖Topology.IsQuotientMap

(root)

Definitions

NameCategoryTheorems
instTopologicalSpaceSubtype 📖CompOp
1309 mathmath: Unitary.openPartialHomeomorph_source, continuous_decomposeProdAdjoint, NonUnitalSubring.instIsTopologicalRing, HasStrictFDerivAt.map_implicitFunctionOfComplemented_eq, TopologicalSpace.IsOpenCover.isInducing_iff_restrictPreimage, continuousSMul_sphere_ball, instCompactSpaceElemClosedBallOfProperSpace, StarModule.decomposeProdAdjointL_symm_apply, ContinuousMap.tendsto_compactOpen_restrict, cfcₙ_def, MeasureTheory.L1.SimpleFunc.norm_Integral_le_one, AddSubmonoid.continuousVAdd, continuous_cfcₙHomSuperset_left, MeromorphicOn.codiscrete_setOf_meromorphicOrderAt_eq_zero_or_top, continuousOn_stereoToFun, Submodule.coe_continuous_linearProjOfClosedCompl', completelyRegularSpace_iff_isOpen, norm_cfcₙHom, IsSeparatedMap.pullback, TopCat.pullbackIsoProdSubtype_hom_fst, OrthonormalBasis.orthogonalProjection_eq_sum, TopologicalSpace.Opens.inclusion'_hom_apply, polynomialFunctions.eq_adjoin_X, HasStrictFDerivAt.implicitToOpenPartialHomeomorphOfComplemented_fst, bernsteinApproximation.apply_zero, Affine.Simplex.orthogonalProjectionSpan_map, ContinuousMap.compactOpen_eq_iInf_induced, EuclideanGeometry.vsub_orthogonalProjection_mem_direction_orthogonal, Matrix.IsHermitian.isClosedEmbedding_cfcAux, IsDiscrete.to_subtype, IsSelfAdjoint.commute_cfcHom, instIsManifoldRealEuclideanSpaceFinModelWithCornersSelfTopWithTopENatElemHAddNatOfNatSphere, NonUnitalIsometricContinuousFunctionalCalculus.isometric, unitInterval.symmHomeomorph_apply, cfcₙHom_apply_mem_elemental, MeasureTheory.L1.SimpleFunc.setToL1SCLM_add_left, OpenPartialHomeomorph.map_subtype_source, Trivialization.preimageHomeomorph_symm_apply, Path.continuousMapClass, ModelWithCorners.interior_open, Subgroup.subgroupOfContinuousMulEquivOfLe_toMulEquiv, nhds_ne_subtype_neBot_iff, ContinuousMap.HomotopyWith.coe_toContinuousMap, OpenPartialHomeomorph.subtypeRestr_symm_trans_subtypeRestr, TopologicalSpace.Opens.instIsManifoldSubtypeMem, instCompletelyRegularSpaceSubtype, AlgebraicGeometry.Scheme.Hom.fiberHomeo_apply, Trivialization.domExtend_source, Topology.IsClosedEmbedding.inclusion, mem_nhdsSet_subtype_iff_nhdsSetWithin, Profinite.NobelingProof.injective_πs', IsClosedMap.isEvenlyCovered_of_openPartialHomeomorph, continuousSMul_sphere_sphere, spectrum.instCompactSpace, Affine.Simplex.orthogonalProjection_vadd_smul_vsub_orthogonalProjection, nhds_ne_subtype_eq_bot_iff, TopologicalSpace.Opens.chartAt_eq, MeasureTheory.norm_condExpL2_le, Metric.contractibleSpace_ball, discreteTopology_subtype_iff', ContinuousLinearMap.ker_codRestrict, Complex.isCoveringMap_exp, ContinuousMap.attachBound_apply_coe, continuousOn_boolIndicator_iff_isClopen, OpenPartialHomeomorph.subtypeRestr_coe, Submodule.re_inner_starProjection_eq_normSq, GenLoop.fromLoop_apply, MeasureTheory.inner_condExpL2_left_eq_right, Unitary.mem_pathComponentOne_iff, ODE.FunSpace.isUniformInducing_toContinuousMap, IsProperMap.restrict, ContinuousFunctionalCalculus.compactSpace_spectrum, IsOpen.isOpenEmbedding_subtypeVal, Trivialization.codExtend_source, EuclideanGeometry.orthogonalProjection_congr, stdSimplexHomeomorphUnitInterval_symm_apply_coe, ContinuousLinearEquiv.snd_equivOfRightInverse, Profinite.NobelingProof.Products.span_nil_eq_top, QuasispectrumRestricts.compactSpace, AlgebraicGeometry.ProjectiveSpectrum.Proj.toSpec_preimage_basicOpen, MeasureTheory.condExpL2_ae_eq_zero_of_ae_eq_zero, ContinuousMapZero.toContinuousMap_id, PiNat.exists_retraction_subtype_of_isClosed, AddSubgroup.addSubgroupOfContinuousAddEquivOfLe_apply, isClosedEmbedding_cfcₙAux, TopologicalSpace.Opens.chartAt_subtype_val_symm_eventuallyEq, ContinuousMapZero.id_toFun, MeasureTheory.Measure.toSphere_apply_aux, unitInterval.toNNReal_continuous, ContinuousLinearEquiv.coe_toSpanNonzeroSingleton_symm, Profinite.NobelingProof.GoodProducts.linearIndependent_iff_union_smaller, Topology.IsInducing.restrictPreimage, Rat.HeightOneSpectrum.adicCompletionIntegers.coe_padicIntEquiv_symm_apply, Submodule.orthogonalProjection_mem_subspace_eq_self, Profinite.NobelingProof.coe_πs, ContinuousMap.Homotopy.apply_zero_path, selfAdjoint.expUnitaryPathToOne_apply, Unitary.continuousOn_argSelfAdjoint, HomotopyGroup.symmAt_indep, Valued.integer.compactSpace_iff_completeSpace_and_isDiscreteValuationRing_and_finite_residueField, Set.instTietzeExtensionUnitBall, TopologicalSpace.noetherian_univ_iff, NonUnitalStarAlgebra.elemental.isClosedEmbedding_coe, contMDiff_neg_sphere, StarAlgebra.elemental.characterSpaceToSpectrum_coe, ContinuousMap.HomotopyWith.prop', ODE.FunSpace.range_toContinuousMap, MeasureTheory.L1.SimpleFunc.setToL1SCLM_const, StarConvex.contractibleSpace, AlgebraicGeometry.ProjIsoSpecTopComponent.toSpec_hom_apply_asIdeal, Path.Homotopy.map_apply, EuclideanGeometry.orthogonalProjection_contLinear, NonUnitalAlgebra.elemental.isClosedEmbedding_coe, Pretrivialization.restrictPreimage_target, Flow.orbit_restrict, EuclideanGeometry.dist_orthogonalProjection_eq_iff_oangle_eq, isCompact_iff_compactSpace, Finset.continuous_restrict_apply, Subtype.isCompact_iff, Profinite.NobelingProof.instSubsingletonLocallyConstantElemForallBoolEmptyCollectionSetInt, AffineSubspace.coe_subtypeA, isTopologicalBasis_subtype, ContinuousMap.Homotopy.evalAt_eq, isLindelof_iff_isLindelof_univ, Topology.IsEmbedding.inclusion, PrimitiveSpectrum.isTopologicalBasis_relativeLower, Subtype.dense_iff, isLindelof_iff_lindelofSpace, instContinuousNegElemBallOfNat, MeasureTheory.L1.norm_setToL1_le_norm_setToL1SCLM, equicontinuous_restrict_iff, polynomialFunctions_separatesPoints, Set.restrictPreimage_isClosedEmbedding, Topology.IsInducing.codRestrict, Submodule.coe_subtypeL', instIsManifoldRealEuclideanSpaceFinOfNatNatEuclideanHalfSpaceModelWithCornersEuclideanHalfSpaceElemIcc, Metric.unitBall.instContinuousMul, PrimeSpectrum.instQuasiSoberElemZeroLocus, contMDiff_inclusion, Profinite.NobelingProof.Products.max_eq_eval, Manifold.IsImmersionAtOfComplement.ofOpen, Metric.contractibleSpace_closedBall, cfcHom_continuous, polynomialFunctions.starClosure_topologicalClosure, cfcL_apply, isPreconnected_iff_preconnectedSpace, isPreirreducible_iff_preirreducibleSpace, Profinite.NobelingProof.Products.eval_πs, Icc_isBoundaryPoint_bot, LinearMap.IsSymmetric.directSum_decompose_apply, Topology.IsLocallyConstructible.iff_of_isOpenCover, cfcHom_nonneg_iff, Homeomorph.contDiff_unitBall, EuclideanGeometry.reflection_vadd_smul_vsub_orthogonalProjection, gelfandStarTransform_naturality, IsOpenMap.restrictPreimage, gelfandStarTransform_apply_apply, TopologicalSpace.Fiber.sigmaIsoHom_inj, StarModule.decomposeProdAdjointL_apply, AddSubgroup.discreteTopology_iff_of_isFiniteRelIndex, Profinite.NobelingProof.GoodProducts.injective, stereographic'_source, cfc_apply_mkD, IsClosed.isCompletelyPseudoMetrizableSpace, EuclideanGeometry.inter_eq_singleton_orthogonalProjection, cfcₙ_eq_cfcₙL_mkD, stdSimplexHomeomorphUnitInterval_zero, continuous_subtype_val, Subgroup.subgroupOf_isOpen, GenLoop.homotopyTo_apply, continuousAt_subtype_val, Profinite.NobelingProof.spanFunctorIsoIndexFunctor_hom_app_hom_hom_apply_coe, ZLattice.comap_discreteTopology, Path.Homotopy.continuous_transAssocReparamAux, MeasureTheory.Lp.simpleFunc.isDenseEmbedding, Set.restrictPreimage_isOpenEmbedding, Subtype.opensMeasurableSpace, Affine.Simplex.orthogonalProjectionSpan_eq_point, compl_mem_codiscrete_iff, Path.Homotopy.target, Manifold.IsImmersion.of_opens, MeasureTheory.MemLp.condExpL2_ae_eq_condExp, CompactlyCoherentSpace.isOpen_iff, gelfandTransform_bijective, ContinuousMap.concatCM_left, Algebra.elemental.isClosedEmbedding_coe, IsRetrocompact_iff_isSpectralMap_subtypeVal, Submodule.IsOrtho.orthogonalProjection_comp_subtypeL, EuclideanGeometry.orthogonalProjection_map, cfcHom_isClosedEmbedding, mfderivWithin_projIcc_one, RestrictedProduct.isOpenEmbedding_structureMap, HomotopyGroup.isUnital_auxGroup, totallyDisconnectedSpace_subtype_iff, HasStrictFDerivAt.eq_implicitFunctionOfComplemented, TopCat.GlueData.ofOpenSubsets_toGlueData_t, IsOpenMap.codRestrict, Profinite.NobelingProof.GoodProducts.equiv_toFun_eq_eval, SpectrumRestricts.compactSpace, Submodule.coe_orthogonalDecomposition_symm, MeasureTheory.aestronglyMeasurable_condExpL2, Submodule.orthogonalProjection_bot, Profinite.NobelingProof.Products.eval_πs', Profinite.NobelingProof.GoodProducts.span_sum, Profinite.NobelingProof.GoodProducts.range_equiv_factorization, TopologicalSpace.NonemptyCompacts.toCompactSpace, EuclideanGeometry.angle_orthogonalProjection_self, EuclideanGeometry.orthogonalProjection_apply_mem, TopologicalSpace.IsOpenCover.generalizingMap_iff_comp, Icc_isInteriorPoint_interior, TopCat.pullbackFst_apply, cfcₙ_apply_mkD, PrimeSpectrum.coe_preimageHomeomorphFiber_apply_asIdeal, CompactlyCoherentSpace.isClosed_iff, cfcₙHom_id, ContinuousOn.restrict_mapsTo, EuclideanGeometry.dist_orthogonalProjection_eq_iff_angle_eq, MeasureTheory.condExpL2_indicator_ae_eq_smul, mfderivWithin_comp_projIcc_one, BoundedContinuousFunction.restrict_apply, SimplexCategory.continuous_toTopMap, preimage_coe_mem_nhds_subtype, isLocallyInjective_iff_isOpen_diagonal, WithTop.tendsto_untop, ContinuousMap.Homotopy.ulift_apply, Real.continuous_tan, MeasureTheory.continuous_diracProbaEquiv, Continuous.restrict, GenLoop.loopHomeo_apply, HasStrictFDerivAt.mem_implicitToPartialHomeomorphOfComplemented_target, OpenPartialHomeomorph.homeomorphOfImageSubsetSource_symm_apply, Subtype.instT25Space, IccLeftChart_extend_interior_pos, Profinite.NobelingProof.fin_comap_jointlySurjective, Set.restrictPreimage_isEmbedding, TopCat.pullbackSnd_apply, MeasureTheory.L1.SimpleFunc.setToL1SCLM_zero_left', TopologicalSpace.Subtype.secondCountableTopology, IsCompactOperator.codRestrict, isBigO_norm_Icc_restrict_atTop, cfcHom_le_iff, ModelWithCorners.isInteriorPoint_iff_isInteriorPoint_val, cfcHom_id, instRegularSpaceSubtype, MeasureTheory.integral_condExpL2_eq, IsCompactOperator.restrict', range_cfcₙHom, Profinite.NobelingProof.GoodProducts.spanFin, Profinite.NobelingProof.succ_exact, ContinuousMap.mkD_of_continuousOn, Unitary.openPartialHomeomorph_symm_apply, mem_intrinsicFrontier, isSigmaCompact_iff_isSigmaCompact_univ, instPathConnectedSpaceElemForallRealStdSimplexOfNonempty, OpenPartialHomeomorph.toHomeomorphSourceTarget_apply_coe, Pretrivialization.restrictPreimage'_target, discreteTopology_of_noAccPts, instIsTopologicalGroupSubtypeMemSubmonoidUnitaryOfContinuousMulOfContinuousStar, TopologicalSpace.IsOpenCover.isOpenMap_iff_comp, Profinite.NobelingProof.GoodProducts.smaller_factorization, IsClosedMap.subtype_mk, MeasureTheory.setIntegral_condExpL2_indicator, contMDiff_coe_sphere, ContinuousMap.mkD_of_not_continuousOn, spectrum.gelfandTransform_eq, stereographic_apply_neg, OrthonormalBasis.orthogonalProjection_apply_eq_sum, ModelWithCorners.boundary_open, ContinuousLinearMap.coe_equivRange, Icc_chartedSpaceChartAt_of_le_top, Real.continuous_log, nhdsWithin_eq_map_subtype_coe, TopologicalSpace.Opens.isOpenEmbedding', Profinite.NobelingProof.πs'_apply_apply, TopologicalSpace.Opens.chartAt_inclusion_symm_eventuallyEq, QuasiconvexOn.isPreconnected_preimage_subtype, range_stereographic_symm, WeakDual.CharacterSpace.continuousMapEval_bijective, HasStrictFDerivAt.to_implicitFunctionOfComplemented, EuclideanGeometry.orthogonalProjection_apply', Affine.Simplex.dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq, isOpenEmbedding_stereographic_symm, MeasureTheory.lintegral_nnnorm_condExpL2_indicator_le, TopCat.pullbackIsoProdSubtype_inv_fst_assoc, continuousMap_mem_polynomialFunctions_closure, SimplexCategory.toTop_obj, Homeomorph.subtype_symm_apply_coe, TopologicalSpace.Opens.partialHomeomorphSubtypeCoe_target, HomotopyGroup.transAt_indep, ProfiniteGrp.denseRange_toLimit, isSeparatedMap_iff_isClosedEmbedding, NonUnitalStarSubalgebra.instIsTopologicalSemiring, AddSubmonoid.continuousAdd, EuclideanGeometry.orthogonalProjection_orthogonalProjection_of_le, TopologicalSpace.IsOpenCover.isClosedEmbedding_iff_restrictPreimage, TopologicalSpace.Opens.instHasGroupoid, StronglyLocallyContractibleSpace.contractible_basis, Trivialization.restrictPreimage'_source, HasStrictFDerivAt.mem_implicitToOpenPartialHomeomorphOfComplemented_target, completelyNormalSpace_iff_forall_normalSpace, contractible_subset_basis, instCompletelyNormalSpaceSubtype, IsOpen.polishSpace, continuous_rangeFactorization_iff, Homeomorph.coe_unitBall_apply_zero, EuclideanGeometry.Sphere.isTangent_iff_isTangentAt_orthogonalProjection, TopCat.continuousPrelocal_pred, continuous_decomposeProdAdjoint_symm, gelfandStarTransform_symm_apply, isIrreducible_iff_irreducibleSpace, AddSubsemigroup.continuousAdd, Flow.coe_restrict_apply, SimplexCategory.toTop_map, MeasureTheory.L1.setToL1_eq_setToL1SCLM, isQuotientMap_projIcc, MeasureTheory.norm_condExpL2_coe_le, MeasureTheory.setLIntegral_nnnorm_condExpL2_indicator_le, IntermediateField.continuousSMul, TopologicalSpace.IsOpenCover.isHomeomorph_iff_restrictPreimage, Path.continuous_delayReflRight, GenLoop.instContinuousEval, Path.Homotopy.hcomp_half, Profinite.NobelingProof.Products.eval_eq, Homeomorph.Set.univ_symm_apply_coe, Submodule.orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero, Path.Homotopy.hcomp_apply, Manifold.IsImmersionAt.of_opens, isCoveringMap_zpow, Path.continuous_delayReflLeft, Path.Homotopy.symm_apply, Submodule.range_subtypeL, WeakDual.CharacterSpace.homeoEval_naturality, semicontinuous_restrict_iff, MeasureTheory.L1.SimpleFunc.setToL1S_smul_real, Path.instContinuousEvalElemRealUnitInterval, NumberField.mixedEmbedding.instDiscreteTopologySubtypeMixedSpaceMemSubmoduleIntIdealLattice, Subgroup.instIsTopologicalGroupSubtypeMem, TopologicalSpace.Opens.instSecondCountableOpens, Path.Homotopy.eval_apply, HilbertBasis.hasSum_orthogonalProjection, TopologicalSpace.IsOpenCover.isClosed_iff_coe_preimage, ContinuousMap.coe_restrict, Unitary.instLocPathConnectedSpace, ContinuousMap.Homotopy.apply_one_path, isLocalHomeomorph_iff_isOpenEmbedding_restrict, IsClosedMap.restrictPreimage, IsCoveringMap.liftPath_const, Pretrivialization.restrictPreimage_toFun, Subalgebra.valA_apply, ContinuousAt.restrictPreimage, MeasureTheory.lintegral_nnnorm_condExpL2_le, SimplexCategory.toTop₀_map, AffineSubspace.signedInfDist_eq_signedDist_of_mem, ContinuousAlgHom.coe_codRestrict, topologicalKrullDim_subspace_le, cfcₙAux_id, Profinite.NobelingProof.Products.evalFacProps, stereographic'_target, Irrational.instOrderTopologySubtypeReal, ContinuousLinearMap.coe_codRestrict, Affine.Simplex.signedInfDist_apply_self, Trivialization.domExtend_symm_apply, continuousSMul_closedBall_ball, IsClosed.sigmaCompactSpace, Pretrivialization.codExtend_baseSet, MeromorphicOn.isClopen_setOf_meromorphicOrderAt_eq_top, ContinuousMap.HomotopyLike.toContinuousMapClass, AffineSubspace.signedInfDist_eq_signedDist_orthogonalProjection, IsCoveringMapOn.isCoveringMap_restrictPreimage, Pretrivialization.codExtend_target, isometry_cfcₙHom, frequently_nhds_subtype_iff, Submodule.norm_orthogonalProjection_apply_le, StarAlgebra.elemental.bijective_characterSpaceToSpectrum, instPolynormableSpaceSubtypeMemSubmodule, HasStrictFDerivAt.map_implicitFunction_eq, bernstein.variance, PadicInt.coe_adicCompletionIntegersEquiv_symm_apply, EuclideanSpace.instIsManifoldSphere, StarSubalgebra.isClosedEmbedding_inclusion, IsClosedMap.mapsToRestrict, bernstein_apply, closure_subtype, gelfandTransform_map_star, polynomialFunctions.starClosure_eq_adjoin_X, isCompact_iff_isCompact_univ, InfiniteGalois.mulEquivToLimit_symm_continuous, MeasureTheory.StronglyMeasurable.separableSpace_range_union_singleton, selfAdjoint.continuous_expUnitary, Nonneg.val_unitsHomeomorphPos_symm_apply_coe, mem_nhds_subtype, Trivialization.codExtend_apply, HasStrictFDerivAt.implicitToOpenPartialHomeomorphOfComplemented_apply_ker, ContinuousMap.aeStronglyMeasurable_restrict_mkD_restrict_of_uncurry, Affine.Simplex.orthogonalProjectionSpan_congr, StarSubalgebra.instIsTopologicalSemiringSubtypeMem, GenLoop.ext_iff, IsClosed.preimage_val, cfcₙHom_predicate, Topology.IsCoherentWith.isClosed_iff, Metric.PiNatEmbed.separation, HasStrictFDerivAt.implicitToPartialHomeomorph_self, TopologicalSpace.MetrizableSpace.subtype, unitInterval.continuous_sigmoid, Subgroup.discrete_iff_cyclic, TopologicalSpace.IsSeparable.secondCountableTopology, ContinuousLinearMap.equivRange_symm_apply, ContinuousMap.restrictPreimage_apply, InfiniteGalois.isOpen_mulEquivToLimit_image_fixingSubgroup, instLocallyConvexSpaceSubtypeMemSubmodule, IsOpen.locallyConnectedSpace, SimplexCategory.toTop₀_obj, IntermediateField.botContinuousSMul, contMDiff_subtype_coe_Icc, ContinuousFunctionalCalculus.exists_cfc_of_predicate, subtype_inseparable_iff, Profinite.IndexFunctor.surjective_π_app, VAddMemClass.continuousVAdd, TopologicalSpace.IsOpenCover.quasiSober_iff_forall, Icc_chartedSpaceChartAt, IsGδ.of_t2Space_locallyCompactSpace, AffineSubspace.instIsTopologicalAddTorsorSubtypeMemSubmoduleDirection, stereoToFun_apply, IsLocallyClosed.locallyCompactSpace, Profinite.NobelingProof.iso_map_bijective, BoundedContinuousFunction.exists_norm_eq_restrict_eq_of_closed, ContinuousLinearMap.coe_projKerOfRightInverse_apply, AddCircle.continuous_equivIoc_symm, stdSimplex.continuous_map, Pretrivialization.codExtend_source, isSeparatedMap_iff_isClosedMap, TopologicalSpace.Opens.openPartialHomeomorphSubtypeCoe_coe, Trivialization.domExtend_baseSet, Path.source', Submodule.coe_continuous_linearProjOfClosedCompl, TopologicalSpace.Opens.coe_overEquivalence_functor_obj, EuclideanGeometry.orthogonalProjection_subtype, Subtype.irreducibleSpace, Filter.Germ.isConstant_comp_subtype, HasStrictFDerivAt.mem_implicitToOpenPartialHomeomorphOfComplemented_source, Submodule.coe_orthogonalDecomposition, EuclideanGeometry.two_zsmul_oangle_self_orthogonalProjection, cfc_def, isLocallyInjective_iff_isOpenMap, isLindelof_iff_LindelofSpace, denseRange_inclusion_iff, connectedComponentIn_eq_image, GenLoop.continuous_fromLoop, AddSubgroup.discrete_iff_addCyclic, Commute.cfcₙHom, Submodule.norm_subtypeL, Subsemigroup.continuousMul, continuousSMul_sphere_closedBall, QuasilinearOn.isPreconnected_preimage_subtype, MeasureTheory.L1.SimpleFunc.setToL1SCLM_congr_left, Path.reparam_id, Metric.sphere.compactSpace, ContinuousAt.codRestrict, ContinuousLinearMap.coe_rangeRestrict, Submodule.norm_sq_eq_add_norm_sq_projection, Convex.contractibleSpace, Trivialization.restrictPreimage'_baseSet, ContinuousLinearMap.equivRange_symm_toLinearEquiv, Nonneg.val_inv_unitsHomeomorphPos_symm_apply_coe, OpenPartialHomeomorph.isOpenEmbedding_restrict, EuclideanGeometry.orthogonalProjection_vadd_eq_self, Submodule.starProjection_apply, HasStrictFDerivAt.mem_implicitToOpenPartialHomeomorph_target, Stonean.extremallyDisconnected_pullback, Profinite.NobelingProof.Products.eval_πs_image', Topology.IsEmbedding.toPullbackDiag, HasStrictFDerivAt.implicitToOpenPartialHomeomorphOfComplemented_self, GenLoop.toLoop_apply, ODE.FunSpace.toContinuousMap_apply_eq_apply, EuclideanGeometry.oangle_orthogonalProjection_self, TopologicalSpace.Opens.overEquivalence_unitIso_hom_app_left, Algebra.quasiFiniteAt_iff_isOpen_singleton_fiber, Submodule.fst_orthogonalDecomposition_apply, IsClosedMap.subtype_map, Profinite.NobelingProof.factors_prod_eq_basis_of_ne, Subtype.instFrechetUrysohnSpace, Topology.IsEmbedding.restrict, Profinite.NobelingProof.continuous_projRestrict, mfderiv_coe_sphere_injective, EuclideanGeometry.dist_orthogonalProjection_eq_dist_iff_eq_of_mem, JoinedIn.joined_subtype, HomotopyGroup.mul_spec, AffineSubspace.abs_signedInfDist_eq_dist_of_mem_affineSpan_insert, TopologicalSpace.Fiber.sigmaIsoHom_apply, IsEvenlyCovered.toTrivialization_apply, Submodule.inner_orthogonalProjection_eq_of_mem_right, range_cfcₙ_eq_range_cfcₙHom, ContinuousMap.adjoin_id_eq_span_one_union, Profinite.NobelingProof.continuous_CC'₁, StructureGroupoid.LocalInvariantProp.liftProp_subtype_val, iccHomeoI_apply_coe, ContinuousAlgHom.coe_codRestrict_apply, Matrix.IsHermitian.cfcAux_apply, AddSubgroup.continuousVAdd, AddSubgroup.discreteTopology_iff_of_finiteIndex, ContinuousMap.nhds_compactOpen_eq_iInf_nhds_induced, instContinuousNegElemSphereOfNat, Affine.Simplex.orthogonalProjection_circumcenter, GenLoop.fromLoop_symm_toLoop, Pretrivialization.codExtend'_toFun, ContinuousLinearMap.coe_linearMap_equivRange, PadicInt.coe_adicCompletionIntegersEquiv_apply, IsCoveringMapOn.restrictPreimage, Continuous.subtype_mk, TopologicalSpace.Compacts.instCompactSpaceSubtypeMem, Real.continuous_logb, AffineSubspace.subtypeA_toAffineMap, Affine.Simplex.exists_forall_dist_eq_iff_exists_excenterExists_and_eq_excenter, Complex.isAddQuotientCoveringMap_exp, HasStrictFDerivAt.to_implicitFunction, inr_comp_cfcₙHom_eq_cfcₙAux, IsOpenMap.subtype_map, Profinite.NobelingProof.GoodProducts.span, Trivialization.codExtend'_target, isLocalHomeomorphOn_iff_isOpenEmbedding_restrict, Condensed.isoLocallyConstantOfIsColimit_inv, Affine.Simplex.orthogonalProjection_eq_circumcenter_of_exists_dist_eq, Continuous.subtype_coind, TopCat.pullbackIsoProdSubtype_inv_snd_apply, Real.continuous_logb', LinearPMap.adjointDomainMkCLM_apply, RestrictedProduct.nhds_eq_map_structureMap, exists_polynomial_near_continuousMap, Topology.IsClosedEmbedding.subtypeVal, Path.isUniformEmbedding_coe, Profinite.NobelingProof.GoodProducts.linearIndependent_iff_smaller, unitInterval.symmHomeomorph_symm_apply, Subtype.connectedSpace, Homeomorph.image_apply_coe, homeomorphUnitSphereProd_symm_apply_coe, unitInterval.continuous_qRight, Subfield.continuousSMul, TopologicalSpace.IsOpenCover.jacobsonSpace_iff, IsOpen.inter_preimage_val_iff, tendstoLocallyUniformlyOn_iff_tendstoLocallyUniformly_comp_coe, MeasureTheory.integral_condExpL2_eq_of_fin_meas_real, Path.Homotopy.trans_apply, LinearMap.compLeftInverse_apply_of_bdd, Metric.PiNatEmbed.exists_closed_embedding_to_hilbert_cube, TopologicalSpace.Opens.partialHomeomorphSubtypeCoe_coe, nhds_eq_nhds_emetric_ball, ContinuousLinearEquiv.toSpanNonzeroSingleton_symm_apply, unitInterval.instTietzeExtension, Nonneg.unitsHomeomorphPos_apply_coe, Continuous.rangeFactorization, Topology.continuous_reorderRestrictProd, IsOpen.locallyCompactSpace, TopCat.pullbackIsoProdSubtype_inv_fst, nhds_subtype_eq_comap, IsOpenMap.mapsToRestrict, WeakDual.CharacterSpace.compContinuousMap_apply, Topology.IsOpenEmbedding.restrictPreimage, StructureGroupoid.LocalInvariantProp.section_spec, Path.Homotopy.continuous_reflTransSymmAux, Ordinal.accPt_subtype, continuous_cfcHomSuperset_left, EuclideanGeometry.orthogonalProjection_affineSpan_singleton, ContinuousLinearEquiv.ofSubmodule'_apply, Homeomorph.unitBall_apply_coe, continuous_IccExtend_iff, Subtype.t3Space, IccLeftChart_extend_bot, IsSimplyConnected.simplyConnectedSpace, IsClosed.isCompletelyMetrizableSpace, cfcₙHom_isClosedEmbedding, smoothSheaf.obj_eq, Trivialization.codExtend'_apply, Subring.instIsTopologicalRing, Submodule.orthogonalProjection_apply_eq_linearProjOfIsCompl, Path.continuous_uncurry_iff, TopologicalSpace.noetherianSpace_set_iff, MeasureTheory.condExpL2_indicator_of_measurable, HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented_self, nhdsWithin_subtype, EuclideanGeometry.dist_sq_eq_dist_orthogonalProjection_sq_add_dist_orthogonalProjection_sq, bernsteinApproximation.apply_one, Module.isLocallyConstant_rankAtStalk_freeLocus, IsClosed.locallyCompactSpace, Pretrivialization.restrictPreimage'_baseSet, ContinuousMultilinearMap.codRestrict_apply_coe, IsCompactOperator.restrict, cfcₙHom_le_iff, bernstein_nonneg, Submodule.sndL_comp_coe_orthogonalDecomposition, stereographic_neg_apply, MeasureTheory.condExpL2_indicator_eq_toSpanSingleton_comp, MeasureTheory.L1.SimpleFunc.norm_setToL1SCLM_le', Profinite.NobelingProof.GoodProducts.linearIndependent_iff_sum, Real.continuous_log', IsNonarchimedeanLocalField.instCompactSpaceSubtypeMemSubringIntegerValueGroupWithZeroValuation, Manifold.riemannianEDist_def, Unitary.openPartialHomeomorph_target, Topology.image_snd_preimageImageRestrict, HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented_fst, unitary.joined, TopCat.pullbackIsoProdSubtype_hom_snd, MeasureTheory.lintegral_nnnorm_condExpL2_indicator_le_real, Topology.IsEmbedding.toHomeomorph_apply_coe, TopologicalSpace.Opens.overEquivalence_counitIso_inv_app, cfc_eq_cfcL_mkD, lowerHemicontinuous_restrict_iff, stdSimplexHomeomorphUnitInterval_apply_coe, IsOpenMap.subtype_mk, Homeomorph.Set.prod_apply, Metric.sphere.instContinuousMul, Subgroup.instDiscreteTopStrictPeriods, Subgroup.instFiniteQuotientSubtypeMemOpenSubgroupOfIsTopologicalGroupOfCompactSpace, ContinuousLinearEquiv.coord_toSpanNonzeroSingleton, homeomorphSphereProd_symm_apply_coe, MeasureTheory.L1.SimpleFunc.setToL1SCLM_smul_left, ContinuousMap.aeStronglyMeasurable_mkD_restrict_of_uncurry, ContinuousMap.Homotopy.curry_apply, instProperVAddSubtypeMemAddSubgroupOfIsClosedCoe, cfcHomSuperset_apply, instT35SpaceSubtype, Submodule.norm_orthogonalProjection_apply, Pretrivialization.restrictPreimage_source, TopologicalSpace.Opens.overEquivalence_unitIso_inv_app_left, MeasureTheory.L1.SimpleFunc.setToL1SCLM_smul_left', EuclideanGeometry.orthogonalProjection_vadd_smul_vsub_orthogonalProjection, MeasureTheory.Measure.measurePreserving_homeomorphUnitSphereProd, TopCat.pullbackIsoProdSubtype_inv_snd, cfcₙHom_eq_cfcₙ_extend, range_cfc_eq_range_cfcHom, instContinuousNegElemClosedBallOfNat, IsOpen.preimage_val, homeomorphUnitSphereProd_apply_fst_coe, Profinite.NobelingProof.e_mem_of_eq_true, MeasureTheory.L1.setToL1_apply_coeToLp, Affine.Simplex.orthogonalProjectionSpan_faceOpposite_eq_point_rev, ZSpan.instDiscreteTopologySubtypeMemSubmoduleIntSpanRangeCoeBasisRealOfFinite, Topology.IsEmbedding.restrictPreimage, Path.Homotopy.cast_apply, contMDiffWithinAt_comp_projIcc_iff, Trivialization.proj_clift, Profinite.IndexFunctor.map_comp_π_app, AddSubgroup.instIsTopologicalAddGroupSubtypeMem, ContinuousMap.continuous_restrict, EuclideanGeometry.dist_orthogonalProjection_eq_infNndist, Profinite.NobelingProof.πs_apply_apply, IsOpenMap.subtype_coind, TopologicalSpace.Opens.openPartialHomeomorphSubtypeCoe_target, GenLoop.toLoop_apply_coe, NumberField.Units.instDiscrete_unitLattice, instBorelSpaceSubtypeMemOpens, NumberField.mixedEmbedding.euclidean.instDiscreteTopologySubtypeMixedSpaceMemSubmoduleIntIntegerLattice, Profinite.NobelingProof.GoodProducts.finsuppSum_mem_span_eval, ContinuousLinearEquiv.toSpanNonzeroSingleton_coord, Submodule.inner_orthogonalProjection_eq_of_mem_left, Metric.PiNatEmbed.continuous_distDenseSeq, Profinite.NobelingProof.GoodProducts.linearIndependent, instDiscreteTopologySubtype, Path.Homotopy.coeFn_injective, Submodule.toLinearMap_orthogonalProjection_eq_linearProjOfIsCompl, ProperVAdd.isProperMap_vadd_pair_set, Homeomorph.image_symm_apply_coe, Profinite.NobelingProof.GoodProducts.square_commutes, nhds_subtype, cfcₙHom_map_quasispectrum, Manifold.IsImmersionOfComplement.ofOpen, IsOpenMap.restrict, ContinuousOn.continuous_restrict_iff_continuous_uniformOnFun, CompletelyRegularSpace.completely_regular, GenLoop.loopHomeo_symm_apply, Subgroup.continuousSMul, Subgroup.subgroupOfContinuousMulEquivOfLe_apply, EuclideanGeometry.Sphere.dist_orthogonalProjection_eq_radius_iff_isTangent, coord_norm', Homeomorph.Set.prod_symm_apply_coe, Convex.locPathConnectedSpace, Icc_chartedSpaceChartAt_of_top_le, spec_cfcₙAux, NonUnitalSubalgebra.instIsTopologicalRing, Profinite.NobelingProof.spanFinBasis.span, instR0SpaceSubtype, Subalgebra.coe_valA, tendsto_nhdsWithin_iff_subtype, oneTangentSpaceIcc_def, cfcHom_eq_cfc_extend, Real.continuous_inv, Path.continuous, Submodule.range_valA, IccRightChart_extend_top_mem_frontier, TopCat.pullbackIsoProdSubtype_inv_fst_apply, ModelWithCorners.BoundarylessManifold.open, EuclideanGeometry.orthogonalProjection_eq_orthogonalProjection_iff_vsub_mem, IsEvenlyCovered.restrictPreimage, continuous_inclusion, MeasureTheory.L1.SimpleFunc.setToL1SCLM_congr_left', unitInterval.tendsto_sigmoid_atTop, isClosed_preimage_val_coborder, stereographic_target, Rat.HeightOneSpectrum.adicCompletionIntegers.coe_padicIntEquiv_apply, Subgroup.discreteTopology_iff_of_isFiniteRelIndex, EuclideanGeometry.dist_eq_iff_dist_orthogonalProjection_eq, OpenPartialHomeomorph.secondCountableTopology_source, StructureGroupoid.trans_restricted, Subtype.isLindelof_iff, SpectrumRestricts.starAlgHom_apply, PrimitiveSpectrum.closedsGC_closureOperator, instIsTopologicalSemiringSubtypeMemSubalgebra, OpenPartialHomeomorph.homeomorphOfImageSubsetSource_apply, Subtype.t1Space, Trivialization.restrictPreimage'_target, cfcₙHomSuperset_id, IsClosedMap.subtype_coind, Submodule.ker_subtypeL, Trivialization.restrictPreimage'_apply, Complex.isOpenQuotientMap_zpow_compl_zero, Unitary.joined, MeasureTheory.integrable_condExpL2_of_isFiniteMeasure, BoundedContinuousFunction.coe_restrict, EuclideanGeometry.dist_orthogonalProjection_eq_infDist, EuclideanGeometry.orthogonalProjection_mem, Commute.cfcHom, continuous_selfAdjointPart, Affine.Simplex.orthogonalProjectionSpan_restrict, Path.subpathAux_continuous, Manifold.IsImmersionAtOfComplement.of_opens, selfAdjointPartL_apply_coe, nnnorm_cfcₙHom, IsClosed.inter_preimage_val_iff, mdifferentiableWithinAt_comp_projIcc_iff, ContinuousMap.nonUnitalStarAlgebraAdjoin_id_subset_ker_evalStarAlgHom, IsClosed.isClosedMap_subtype_val, QuasiconcaveOn.isPreconnected_preimage_subtype, cfc_apply, HomotopyGroup.one_def, isSigmaCompact_iff_sigmaCompactSpace, ContinuousMap.mkD_apply_of_continuousOn, AnalyticOnNhd.isClopen_setOf_analyticOrderAt_eq_top, Path.target', TopologicalSpace.Opens.openPartialHomeomorphSubtypeCoe_source, Set.instTietzeExtensionUnitClosedBall, WeakDual.metrizable_of_isCompact, MeasureTheory.integrable_condExpL2_indicator, mem_nhds_subtype_iff_nhdsWithin, StructureGroupoid.LocalInvariantProp.liftProp_inclusion, StructureGroupoid.restriction_in_maximalAtlas, EuclideanGeometry.dist_orthogonalProjection_eq_zero_iff, range_mfderiv_coe_sphere, ContinuousMap.Homotopy.extend_of_mem_I, TopologicalSpace.IsOpenCover.isOpen_iff_coe_preimage, IsDenseEmbedding.subtype, PrimitiveSpectrum.isOpen_iff, stereographic_source, HasStrictFDerivAt.implicitToOpenPartialHomeomorph_self, t5Space_iff_forall_t4Space, IsOpen.locPathConnectedSpace, AnalyticOnNhd.codiscrete_setOf_analyticOrderAt_eq_zero_or_top, NonUnitalStarSubalgebra.instIsTopologicalRing, polynomialFunctions_coe, EuclideanGeometry.exists_dist_eq_iff_exists_dist_orthogonalProjection_eq, ContinuousMultilinearMap.codRestrict_toMultilinearMap, isClosedEmbedding_cfcₙHom_of_cfcHom, ContinuousLinearMap.projKerOfRightInverse_comp_inv, continuousOn_iff_continuous_restrict, iccLeftChart_extend_zero, EuclideanGeometry.Sphere.IsTangentAt.eq_orthogonalProjection, Trivialization.domExtend_target, MeasureTheory.MemLp.condExpL2_ae_eq_condExp', Valued.integer.properSpace_iff_compactSpace_integer, Subgroup.discreteTopology_iff_of_finiteIndex, IsSelfAdjoint.commute_cfcₙHom, QuasispectrumRestricts.nonUnitalStarAlgHom_apply, Homeomorph.subtype_toEquiv, AddSubgroup.instDiscreteTopologyZMultiples, ContinuousMap.exists_tendsto_compactOpen_iff_forall, NumberField.mixedEmbedding.instDiscreteTopologySubtypeMixedSpaceMemSubmoduleIntIntegerLattice, IsOpen.baireSpace, isLocallyClosed_tfae, lowerSemicontinuous_restrict_iff, instR1SpaceSubtype, ContinuousMapZero.aeStronglyMeasurable_restrict_mkD_restrict_of_uncurry, HasStrictFDerivAt.implicitToOpenPartialHomeomorph_fst, NonUnitalContinuousFunctionalCalculus.exists_cfc_of_predicate, Subring.continuousSMul, Unitary.isPathConnected_ball, Submodule.subtypeL_apply, stdSimplexHomeomorphUnitInterval_one, Subsemiring.topologicalSemiring, ContinuousMap.Homotopy.continuous, Profinite.NobelingProof.continuous_projRestricts, EuclideanGeometry.reflection_eq_iff_orthogonalProjection_eq, HasStrictFDerivAt.implicitToOpenPartialHomeomorph_apply_ker, Submodule.ker_orthogonalProjection, continuousAt_codRestrict_iff, MeasureTheory.L1.SimpleFunc.integral_smul, IsLocallyInjective_iff_isOpenEmbedding, cfcHom_isStrictlyPositive_iff, Pi.continuous_restrict_apply, instContinuousStarSubtypeMemSubmonoidUnitary, Profinite.NobelingProof.GoodProducts.smaller_mono, SMulMemClass.continuousSMul, EuclideanGeometry.orthogonalProjection_orthogonalProjection, TopCat.pullbackIsoProdSubtype_hom_apply, MeasureTheory.L1.SimpleFunc.norm_setToL1SCLM_le, eventually_nhds_subtype_iff, HomotopyGroup.inv_spec, EuclideanGeometry.orthogonalProjection_vsub_mem_direction, discreteTopology_subtype_iff, IsClosedMap.codRestrict, instContinuousSMulSubtypeLeOfNat, ContinuousOn.restrict, MeasureTheory.L1.SimpleFunc.setToL1SCLM_congr_measure, Pretrivialization.restrictPreimage'_toFun, Path.range_coe, nhdsWithin_subtype_eq_bot_iff, Continuous.codRestrict, separatesPoints_continuous_of_t35Space_Icc, iccHomeoI_symm_apply_coe, Trivialization.preimageHomeomorph_apply, separableSpace_univ, TopologicalSpace.IsSeparable.separableSpace, ContinuousLinearEquiv.submoduleMap_symm_apply, TopologicalSpace.IsOpenCover.denseRange_iff_restrictPreimage, Affine.Simplex.orthogonalProjection_eq_circumcenter_of_dist_eq, StarAlgebra.elemental.isClosedEmbedding_coe, ContinuousMap.norm_restrict_mono_set, MeasureTheory.locallyIntegrable_comap, cfc_apply_pi, Submonoid.continuousSMul, Trivialization.restrictPreimage_source, PeriodPair.instDiscreteTopologySubtypeComplexMemSubmoduleIntLattice, IsLocalHomeomorphOn.discreteTopology_image_iff, Profinite.NobelingProof.GoodProducts.linearIndependentEmpty, ContinuousLinearMap.range_eq_map_coprodSubtypeLEquivOfIsCompl, ContinuousMap.comp_attachBound_mem_closure, nhds_subtype_eq_comap_nhdsWithin, Homeomorph.Set.univ_apply, Subgroup.subgroupOfContinuousMulEquivOfLe_symm_apply, UniformOnFun.edist_continuousRestrict, ZSpan.instDiscreteTopologySubtypeMemAddSubgroupToAddSubgroupIntSpanRangeCoeBasisRealOfFinite, TopologicalSpace.Opens.coe_overEquivalence_inverse_obj_left, EuclideanGeometry.vsub_orthogonalProjection_mem_direction, ContinuousMap.Homotopy.coe_toContinuousMap, Trivialization.preimageSingletonHomeomorph_symm_apply, ContinuousMap.continuousOn_mkD_restrict_of_uncurry, ContinuousMap.tendsto_compactOpen_iff_forall, Matrix.IsHermitian.cfcAux_id, cfcₙAux_mem_range_inr, ContinuousMap.adjoin_id_eq_span_one_add, ContinuousMapZero.hasFiniteIntegral_mkD_restrict_of_bound, OpenPartialHomeomorph.subtypeRestr_symm_eqOn, AddCircle.continuous_equivIco_symm, contMDiffAt_subtype_iff, Path.subpath_continuous_family, Subtype.preirreducibleSpace, IsProperMap.restrictPreimage, Affine.Simplex.coe_orthogonalProjection_vadd_smul_vsub_orthogonalProjection, ContinuousMap.continuous_mkD_restrict_of_uncurry, BoundedContinuousFunction.dist_extend_extend, skewAdjointPartL_apply_coe, ModelWithCorners.isBoundaryPoint_iff_isBoundaryPoint_val, AddCircle.continuousAt_equivIoc, Submonoid.continuousMul, HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented_apply, ProfiniteGrp.instCompactSpaceSubtypeForallCarrierToTopTotallyDisconnectedSpaceToProfiniteObjMemSubgroupLimitConePtAux, Topology.IsEmbedding.homeomorphOfSubsetRange_apply_coe, ContinuousLinearMap.projKerOfRightInverse_apply_idem, Homeomorph.subtype_apply_coe, Set.restrictPreimage_isInducing, Metric.contractibleSpace_eball, IsLocalHomeomorphOn.discreteTopology_of_image, PrimitiveSpectrum.isClosed_iff, Path.Homotopy.continuous_transReflReparamAux, cantorToHilbert_continuous, cfcHomSuperset_id, instT2SpaceSubtype, subtype_specializes_iff, EuclideanGeometry.Sphere.IsTangent.isTangentAt, OpenPartialHomeomorph.subtypeRestr_def, continuous_projIcc, ContinuousLinearMap.coe_codRestrict_apply, NonUnitalSubsemiring.instIsTopologicalSemiring, LightCondSet.LocallyConstant.instHasPropAndTotallyDisconnectedSpaceCarrierSecondCountableTopologySubtypeToTop, ContinuousLinearMap.IsPositive.orthogonalProjection_comp, ContinuousMap.HomotopyWith.prop, TopCat.pullbackIsoProdSubtype_inv_snd_assoc, ContinuousMap.restrict_apply_mk, OpenPartialHomeomorph.subtypeRestr_target_subset, dimH_orthogonalProjection_le, instDiscreteTopologySubtypeMemSubmoduleIntComap, Path.id_apply, isBigO_norm_Icc_restrict_atBot, Int.instDiscreteTopologySubtypeRealMemAddSubgroupZmultiples, DiscreteTopology.of_subset, EuclideanGeometry.reflection_apply', Trivialization.restrictPreimage_baseSet, Profinite.NobelingProof.GoodProducts.span_iff_products, cfcₙHom_continuous, ODE.FunSpace.continuous, EuclideanGeometry.oangle_self_orthogonalProjection, WeakDual.gelfandTransform_apply_apply, Pretrivialization.codExtend'_target, MeasureTheory.tendsto_diracProbaEquivSymm_iff_tendsto, HasStrictFDerivAt.mem_implicitToPartialHomeomorphOfComplemented_source, Topology.IsQuotientMap.restrictPreimage_isOpen, cfcₙHomSuperset_continuous, Profinite.NobelingProof.GoodProducts.sum_equiv_comp_eval_eq_elim, Path.coe_toContinuousMap, OpenPartialHomeomorph.subtypeRestr_symm_eqOn_of_le, bernsteinApproximation_uniform, ContinuousMap.Homotopy.map_one_left, AddCircle.continuousAt_equivIco, Pretrivialization.restrictPreimage'_source, continuousSMul_closedBall_closedBall, Subtype.instOrderClosedTopology, cfcₙ_apply_pi, Homeomorph.ofEqSubtypes_toEquiv, instProperSMulSubtypeMulOppositeMemSubgroupOpOfIsTopologicalGroupOfIsClosedCoe, TopologicalSpace.Fiber.sigmaIsoHom_surj, isCoveringMap_npow, MeasureTheory.L1.SimpleFunc.setToL1SCLM_mono_left', ContinuousMapZero.mkD_of_not_continuousOn, ContinuousMap.hasFiniteIntegral_mkD_restrict_of_bound, ContinuousAlgHom.coe_rangeRestrict, ContinuousLinearMap.exist_extension_of_finiteDimensional_range, MeasureTheory.hausdorffMeasure_orthogonalProjection_le, GenLoop.Homotopic.equiv, ContinuousMap.injective_restrict, contMDiffOn_comp_projIcc_iff, Trivialization.sourceHomeomorphBaseSetProd_symm_apply, Path.truncate_const_continuous_family, IsOpen.isOpenMap_inclusion, Submodule.lipschitzWith_orthogonalProjection, Affine.Simplex.abs_signedInfDist_eq_dist_of_mem_affineSpan_range, NNReal.summable_mk, isPathConnected_iff_pathConnectedSpace, isClopen_preimage_val, TopologicalSpace.NoetherianSpace.set, TopologicalSpace.IsOpenCover.isEmbedding_iff_restrictPreimage, ContinuousLinearEquiv.submoduleMap_apply, Submodule.ClosedComplemented.exists_submodule_equiv_prod, Manifold.IsImmersion.ofOpen, continuousWithinAt_iff_continuousAt_restrict, Affine.Simplex.orthogonalProjectionSpan_reindex, Topology.IsClosedEmbedding.restrictPreimage, NonUnitalContinuousFunctionalCalculus.compactSpace_quasispectrum, ContinuousLinearMap.tendsto_birkhoffAverage_orthogonalProjection, cfc_eq_cfcL, stereographic_apply, MeasureTheory.inner_condExpL2_eq_inner_fun, homeomorphSphereProd_apply_snd_coe, MeasureTheory.integrableOn_condExpL2_of_measure_ne_top, smoothSheaf.contMDiff_section, Metric.eball_contractible, Topology.IsOpenEmbedding.restrict, isDiscrete_iff_discreteTopology, polynomialFunctions.topologicalClosure, Submodule.topologicalAddGroup, NonUnitalSubalgebra.instIsTopologicalSemiring, Submodule.fstL_comp_coe_orthogonalDecomposition, tendsto_subtype_rng, ContinuousMap.concatCM_right, isClosed_preimage_val, MeasureTheory.L1.SimpleFunc.setToL1SCLM_zero_left, equicontinuousAt_restrict_iff, Subtype.preconnectedSpace, Complex.continuous_tan, ProperSMul.isProperMap_smul_pair_set, EuclideanGeometry.orthogonalProjection_linear, Flow.restrictAddSubmonoid_apply, ContinuousLinearEquiv.ofSubmodule'_symm_apply, ContinuousMapZero.aeStronglyMeasurable_mkD_restrict_of_uncurry, cfcHom_predicate, Submodule.orthogonalProjection_orthogonal_apply_eq_zero, AffineSubspace.signedInfDist_def, WeakDual.CharacterSpace.continuousMapEval_apply_apply, IsClosed.weaklyLocallyCompactSpace, Path.truncate_continuous_family, StructureGroupoid.LocalInvariantProp.liftPropAt_iff_comp_subtype_val, TopologicalSpace.Fiber.instCompactSpaceElemValSetMemRangeCoeLocallyConstantPreimageSingleton, exists_extension_norm_eq, ContinuousOn.mapsToRestrict, spectrum.instCompactSpaceNNReal, Profinite.NobelingProof.CC_comp_zero, exists_infinite_discreteTopology, Dense.isDenseEmbedding_val, IsClosed.isClosedEmbedding_subtypeVal, Metric.contractibleSpace_closedEBall, Topology.IsEmbedding.codRestrict, IsLocalHomeomorph.discreteTopology_range_iff, ContinuousMap.IccExtendCM_of_mem, Affine.Simplex.orthogonalProjectionSpan_eulerPoint_mem_ninePointCircle, Manifold.IsImmersionAt.ofOpen, Manifold.lintegral_norm_mfderiv_Icc_eq_pathELength_projIcc, Profinite.NobelingProof.eval_eq_πJ, AddSubgroup.Commensurable.discreteTopology_iff, ContinuousMap.polynomial_comp_attachBound, Profinite.NobelingProof.continuous_CC'₀, IsPathConnected.preimage_coe, norm_cfcHom, IsClosedMap.restrict, Profinite.NobelingProof.Products.evalFacProp, PrimeSpectrum.coe_preimageHomeomorphFiber_symm_apply_coe_asIdeal, Submodule.orthogonalProjectionFn_eq, Complex.isOpenQuotientMap_pow_compl_zero, ZSpan.discreteTopology_pi_basisFun, Affine.Simplex.signedInfDist_affineCombination, instProperSMulSubtypeMemSubgroupOfIsClosedCoe, ContinuousMap.HomotopyWith.continuous, EuclideanGeometry.angle_self_orthogonalProjection, homeomorphUnitSphereProd_apply_snd_coe, IsLocalHomeomorph.isTopologicalBasis, Profinite.NobelingProof.Products.evalCons, ContinuousAlternatingMap.codRestrict_apply_coe, TopologicalSpace.Opens.isOpenEmbedding_of_le, GeneralizingMap.restrictPreimage, Trivialization.codExtend_target, TopologicalSpace.IsOpenCover.isLocallyClosed_iff_coe_preimage, Pretrivialization.restrictPreimage_baseSet, MeasureTheory.continuous_diracProbaEquivSymm, instContinuousInvSubtypeMemSubmonoidUnitaryOfContinuousStar, IsGδ.baireSpace_of_dense, IsEvenlyCovered.of_openPartialHomeomorph, EuclideanGeometry.orthogonalProjection_eq_self_iff, Cube.insertAt_boundary, ContinuousMap.ker_evalStarAlgHom_inter_adjoin_id, cfcₙ_eq_cfcₙL, Profinite.NobelingProof.one_sub_e_mem_of_false, PadicInt.isOpenEmbedding_coe, ContinuousLinearEquiv.coord_norm', ProfiniteGrp.toLimitFun_continuous, Subtype.isSigmaCompact_iff, Subalgebra.continuousSMul, stereographic'_symm_apply, ContinuousOn.tendsto_restrict_iff_tendstoUniformlyOn, cfcₙL_apply, homeomorphSphereProd_apply_fst_coe, Set.Countable.totallySeparatedSpace, Submodule.snd_orthogonalDecomposition_apply, cfcHomSuperset_continuous, AddSubgroup.addSubgroupOfContinuousAddEquivOfLe_symm_apply, instT5SpaceSubtype, MeasureTheory.Lp.compMeasurePreserving_continuous, Metric.unitClosedBall.instContinuousMul, discreteTopology_union, InfiniteGalois.algEquivToLimit_continuous, ContinuousMap.elemental_id_eq_top, gelfandTransform_isometry, Path.Homotopy.source, Profinite.NobelingProof.injective_πs, WeakDual.CharacterSpace.compContinuousMap_comp, TopologicalSpace.Subtype.firstCountableTopology, polynomialFunctions_closure_eq_top', Subtype.borelSpace, bernstein.probability, completelyRegularSpace_iff, Submodule.orthogonalDecomposition_apply, unitary.isPathConnected_ball, ClosedSubgroup.instCompactSpaceSubtypeMem, ContinuousMap.polynomial_comp_attachBound_mem, range_cfcHom, Matrix.SpecialLinearGroup.discreteSpecialLinearGroupIntRange, ContinuousMapZero.mkD_of_continuousOn, MeasureTheory.L1.SimpleFunc.setToL1SCLM_mono_left, Subsemiring.continuousSMul, HasStrictFDerivAt.mem_implicitToPartialHomeomorph_source, nnnorm_cfcHom, Profinite.NobelingProof.coe_πs', TopologicalSpace.Opens.overEquivalence_inverse_obj_right_as, HasStrictFDerivAt.implicitToPartialHomeomorph_fst, HasStrictFDerivAt.mem_implicitToPartialHomeomorph_target, EuclideanGeometry.orthogonalProjection_vsub_mem_direction_orthogonal, Profinite.NobelingProof.GoodProducts.range_equiv_smaller_toFun_bijective, MeasureTheory.eLpNorm_condExpL2_le, IsEvenlyCovered.mem_toTrivialization_baseSet, ContinuousLinearEquiv.toSpanNonzeroSingleton_apply_coe, exists_compact_surjective_zorn_subset, ContinuousLinearEquiv.coord_self, compactSpace_Icc, Trivialization.sourceHomeomorphBaseSetProd_apply, UniformOnFun.edist_continuousRestrict_of_singleton, Manifold.IsImmersionOfComplement.of_opens, DiscreteTopology.preimage_of_continuous_injective, ContinuousLinearEquiv.coord_norm, StarAlgebra.elemental.continuous_characterSpaceToSpectrum, HasStrictFDerivAt.implicitToPartialHomeomorphOfComplemented_apply_ker, RestrictedProduct.nhds_zero_eq_map_structureMap, LinearPMap.mem_adjoint_domain_iff, AddSubgroup.addSubgroupOfContinuousAddEquivOfLe_toAddEquiv, map_nhds_subtype_coe_eq_nhds, MeasureTheory.Lp.simpleFunc.denseRange_coeSimpleFuncNonnegToLpNonneg, EuclideanGeometry.eq_orthogonalProjection_of_eq_subspace, EuclideanGeometry.orthogonalProjection_apply, ContinuousMapZero.elemental_eq_top, ContinuousMapZero.adjoin_id_dense, MeasureTheory.L1.SimpleFunc.setToL1SCLM_mono, t5Space_iff_forall_isOpen_t4Space, Submodule.norm_orthogonalProjection, AffineSubspace.signedInfDist_apply_self, EuclideanGeometry.orthogonalProjection_vsub_orthogonalProjection, ContinuousMap.compactOpen_le_induced, EuclideanGeometry.orthogonalProjection_mem_orthogonal, Topology.IsEmbedding.subtypeVal, ContinuousMap.ker_evalStarAlgHom_eq_closure_adjoin_id, ContMDiff.codRestrict_sphere, Subtype.instAlexandrovDiscrete, completelyNormalSpace_iff_forall_isOpen_normalSpace, Profinite.NobelingProof.list_prod_apply, ContinuousMapZero.mkD_apply_of_continuousOn, Topology.IsEmbedding.toHomeomorph_symm_apply, MeasureTheory.condExpIndSMul_ae_eq_smul, Topology.isEmbedding_sigmoid, MeasureTheory.Lp.simpleFunc.isDenseInducing, IsClosed.isProperMap_subtypeVal, IsClosed.polishSpace, instProperVAddSubtypeAddOppositeMemAddSubgroupOpOfIsTopologicalAddGroupOfIsClosedCoe, Subtype.t0Space, Path.Homotopy.symm₂_apply, Metric.instTietzeExtensionClosedBall, TopologicalSpace.IsOpenCover.isOpenEmbedding_iff_restrictPreimage, EuclideanGeometry.two_zsmul_oangle_orthogonalProjection_self, MeasureTheory.L1.setToL1'_eq_setToL1SCLM, Trivialization.restrictPreimage_apply, mfderiv_subtype_coe_Icc_one, unitInterval.tendsto_sigmoid_atBot, Pretrivialization.codExtend_toFun, TopologicalSpace.IsOpenCover.isClosedMap_iff_restrictPreimage, BoundedContinuousFunction.exists_norm_eq_restrict_eq, mem_intrinsicInterior, Continuous.subtype_map, IsOpen.isOpenMap_subtype_val, IsometricContinuousFunctionalCalculus.isometric, StructureGroupoid.subtypeRestr_mem_maximalAtlas, ContinuousMap.instCompactSpaceElemCoeCompacts, Trivialization.codExtend'_baseSet, Subtype.totallyDisconnectedSpace, ContinuousMap.Homotopy.curry_one, upperHemicontinuousOn_iff_restrict, AddSubgroup.addSubgroupOf_isOpen, Unitary.openPartialHomeomorph_apply, MeasureTheory.L1.SimpleFunc.setToL1SCLM_nonneg, TopologicalSpace.Opens.overEquivalence_counitIso_hom_app, SetLike.isDiscrete_iff_discreteTopology, StructureGroupoid.LocalInvariantProp.liftPropAt_iff_comp_inclusion, Submodule.orthogonalProjection_comp_subtypeL_eq_zero_iff, WithTop.continuous_untop, Metric.instTietzeExtensionBall, isQuasiSeparated_iff_quasiSeparatedSpace, quasispectrum.instCompactSpaceNNReal, cfcₙHom_of_cfcHom_map_quasispectrum, EuclideanGeometry.coe_orthogonalProjection_eq_iff_mem, WeakDual.CharacterSpace.compContinuousMap_id, Manifold.IsSmoothEmbedding.of_opens, OpenPartialHomeomorph.subtypeRestr_source, TopologicalSpace.PseudoMetrizableSpace.subtype, Pretrivialization.codExtend'_source, Profinite.NobelingProof.GoodProducts.linearIndependent_iff_range, Polynomial.toContinuousMapOn_apply, continuous_stereoInvFun, IsEvenlyCovered.discreteTopology_fiber, Submodule.orthogonalProjection_eq_zero_iff, Trivialization.codExtend'_source, Profinite.NobelingProof.GoodProducts.max_eq_eval, polynomialFunctions_closure_eq_top, IsLocallyClosed.isOpen_preimage_val_closure, cfcₙHomSuperset_apply, Submodule.orthogonalProjection_starProjection_of_le, Real.exists_extension_norm_eq, cfcₙ_apply, Submodule.coe_orthogonalProjection_apply, Subgroup.instDiscreteTopologyZMultiples, IsClosed.isClosedMap_inclusion, Stonean.extremallyDisconnected_preimage, IsEvenlyCovered.to_isEvenlyCovered_preimage, cfcHom_eq_of_isStarNormal, continuous_skewAdjointPart, Path.Homotopy.refl_apply, Trivialization.codExtend_baseSet, Submodule.norm_subtypeL_le, WeakDual.CharacterSpace.instCompactSpaceElemCharacterSpaceOfProperSpace, ContinuousLinearEquiv.fst_equivOfRightInverse, cfcHom_map_spectrum, mem_nhdsWithin_subtype, upperSemicontinuousOn_iff_restrict, AlgebraicGeometry.Scheme.Hom.fiberι_fiberHomeo_symm, GenLoop.boundary, IccRightChart_extend_top, GenLoop.homotopyFrom_apply, GenLoop.const_apply, Pretrivialization.codExtend'_baseSet, Profinite.NobelingProof.GoodProducts.linearIndependentSingleton, stdSimplex.instCompactSpace_coe, EuclideanGeometry.dist_orthogonalProjection_line_eq_iff_two_zsmul_oangle_eq, Continuous.restrictPreimage, orderTopology_of_ordConnected, EuclideanGeometry.orthogonalProjection_eq_iff_mem, IsCoveringMap.restrictPreimage, unitary.mem_pathComponentOne_iff, EuclideanGeometry.dist_orthogonalProjection_line_eq_of_two_zsmul_oangle_eq, boundary_Icc, Submodule.coe_subtypeL, HasStrictFDerivAt.implicitToPartialHomeomorph_apply_ker, Submodule.orthogonalProjection_coe_eq_linearProjOfIsCompl, Subgroup.instDiscreteTopPeriods, MeasureTheory.L1.setToL1'_apply_coeToLp, StarSubalgebra.isEmbedding_inclusion, Subgroup.Commensurable.discreteTopology_iff, cfcₙHom_nonneg_iff, NormedSpace.discreteTopology_zmultiples, GenLoop.fromLoop_coe, Profinite.NobelingProof.Products.eval_πs_image, MeasureTheory.Measure.toSphere.instIsOpenPosMeasure, Subgroup.IsArithmetic.discreteTopology, Submodule.orthogonalProjection_norm_le, ContinuousLinearEquiv.equivOfRightInverse_symm_apply, ProfiniteAddGrp.instCompactSpaceSubtypeForallCarrierToTopTotallyDisconnectedSpaceToProfiniteObjMemAddSubgroupLimitConePtAux, TopologicalSpace.IsCompletelyMetrizableSpace.univ, GenLoop.homotopicTo, Submodule.orthogonalProjection_orthogonal, continuousFunctionalCalculus_map_id, boundary_product, MeasureTheory.norm_condExpL2_le_one, isBigO_norm_restrict_cocompact, TopologicalSpace.Opens.partialHomeomorphSubtypeCoe_source, GenLoop.continuous_toLoop, MeasureTheory.condExpL2_comp_continuousLinearMap, Trivialization.preimageSingletonHomeomorph_apply, Submodule.orthogonalProjection_orthogonalComplement_singleton_eq_zero, GenLoop.fromLoop_trans_toLoop, Dense.isDenseInducing_val, Subalgebra.coe_valA', EuclideanGeometry.Sphere.dist_orthogonalProjection_eq_radius_iff_isTangentAt, Real.fromBinary_continuous, ClosedAddSubgroup.instCompactSpaceSubtypeMem, Submodule.orthogonalProjection_eq_linearProjOfIsCompl, joinedIn_iff_joined, CantorScheme.VanishingDiam.map_continuous, MeasureTheory.condExpL2_const_inner, Unitary.path_apply, Topology.IsInducing.subtypeVal, toTopologicalSpace_subtype, Polynomial.toContinuousMapOnAlgHom_apply, mem_codiscrete_subtype_iff_mem_codiscreteWithin, Profinite.NobelingProof.factors_prod_eq_basis, Submodule.coe_prodEquivOfClosedCompl, isSeparatedMap_iff_isClosed_diagonal, EuclideanGeometry.dist_set_eq_iff_dist_orthogonalProjection_eq, ContinuousMap.Homotopy.map_zero_left, unitInterval.continuous_symm, HasStrictFDerivAt.mem_implicitToOpenPartialHomeomorph_source, StructureGroupoid.restriction_mem_maximalAtlas_subtype, HasStrictFDerivAt.eq_implicitFunction, IccLeftChart_extend_bot_mem_frontier, cfcHom_apply_mem_elemental, ContinuousLinearEquiv.ofSubmodule'_toContinuousLinearMap, Icc_isBoundaryPoint_top, instDiscreteTopologySubtypeGeneralLinearGroupMemSubgroupAdjoinNegOneOfIsTopologicalRingOfT2Space, LinearMap.exists_map_addHaar_eq_smul_addHaar', Profinite.NobelingProof.factors_prod_eq_basis_of_eq, ContinuousMap.coe_IccExtend, Profinite.NobelingProof.GoodProducts.union, ContinuousAt.restrict, AddSubgroup.instFiniteQuotientSubtypeMemOpenAddSubgroupOfIsTopologicalAddGroupOfCompactSpace, map_nhds_subtype_val, IsOpen.stronglyLocallyContractibleSpace, OrthogonalFamily.projection_directSum_coeAddHom, TopologicalSpace.IsOpenCover.generalizingMap_iff_restrictPreimage, surjective_stereographic, contMDiff_subtype_val, hasFDerivAt_stereoInvFunAux_comp_coe, MeasureTheory.condExpL2_indicator_nonneg, Submodule.coe_prodEquivOfClosedCompl_symm, instIsManifoldIcc, OpenPartialHomeomorph.toHomeomorphSourceTarget_symm_apply_coe, GenLoop.instContinuousEvalConst, CompletelyRegularSpace.completely_regular_isOpen, ContinuousLinearEquiv.ofSubmodules_apply, contMDiffOn_projIcc, MeasureTheory.L1.SimpleFunc.setToL1SCLM_add_left', TopologicalSpace.IsOpenCover.isOpenMap_iff_restrictPreimage, Continuous.mapPullback, RestrictedProduct.isEmbedding_structureMap, Topology.IsCoherentWith.isOpen_iff, instHereditarilyLindelofSpaceSubtype, map_nhdsSet_subtype_val, bernsteinApproximation.apply, quasispectrum.instCompactSpace, mem_intrinsicClosure, isometry_cfcHom, ApproximatesLinearOn.continuous, ContinuousMap.Homotopy.curry_zero, Metric.sphere.instIsTopologicalGroup, Metric.PiNatEmbed.exists_embedding_to_hilbert_cube, selfAdjoint.joined_one_expUnitary, EuclideanGeometry.orthogonalProjection_mem_subspace_eq_self, Profinite.NobelingProof.GoodProducts.max_eq_eval_unapply, HasStrictFDerivAt.implicitToOpenPartialHomeomorphOfComplemented_apply, polynomialFunctions.comap_compRightAlgHom_iccHomeoI, Homeomorph.unitBall_symm_apply, Metric.ball_contractible, unitary.continuousOn_argSelfAdjoint, Profinite.NobelingProof.succ_mono, Trivialization.restrictPreimage_target, MeasureTheory.L1.SimpleFunc.setToL1S_smul, isConnected_iff_connectedSpace, ContinuousLinearEquiv.ofSubmodules_symm_apply, unitInterval.instConnectedSpaceElemReal, IsCoveringMap.liftPath_trans, TopologicalSpace.NoetherianSpace.range, ContinuousMap.restrict_apply, isSimplyConnected_iff_exists_homotopy_refl_forall_mem

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