Documentation Verification Report

Continuous

📁 Source: Mathlib/Topology/Continuous.lean

Statistics

MetricCount
DefinitionsContinuous, some, comp_of_eq_lemmas, continuity_lemma_statement
4
Theoremsmap, closure_preimage_subset, comp, comp', continuousAt, frontier_preimage_subset, iterate, range_subset_closure_image_dense, tendsto, tendsto', comp, comp', comp_of_eq, eventually_mem, iterate, preimage_mem_nhds, tendsto, denseRange_val, closure_range, comp, dense_image, dense_of_mapsTo, exists_mem_open, mem_nhds, nonempty, nonempty_iff, subset_closure_image_preimage_of_isOpen, continuous_symm_iff, isOpenMap_symm_iff, continuousAt, lift'_closure, denseRange, preimage, preimage, closure, closure_left, closure_image_closure, closure_subset_preimage_closure_image, continuousAt_congr, continuousAt_const, continuousAt_def, continuousAt_id, continuousAt_id', continuous_congr, continuous_const, continuous_def, continuous_id, continuous_id', continuous_iff_continuousAt, continuous_iff_image_closure_subset_closure_image, continuous_iff_isClosed, continuous_iff_preimage_interior_subset_interior_preimage, continuous_of_const, denseRange_id, denseRange_iff_closure_range, denseRange_subtype_val, image_closure_subset_closure_image, map_mem_closure, mem_closure_image, not_continuousAt_of_tendsto, preimage_interior_subset_interior_preimage, tendsto_lift'_closure_nhds
62
Total66

ClusterPt

Theorems

NameKindAssumesProvesValidatesDepends On
map 📖mathematicalContinuousAt
Filter.Tendsto
ClusterPtFilter.NeBot.mono
Filter.NeBot.map
Filter.Tendsto.inf
ContinuousAt.tendsto

Continuous

Theorems

NameKindAssumesProvesValidatesDepends On
closure_preimage_subset 📖mathematicalContinuousSet
Set.instHasSubset
closure
Set.preimage
IsClosed.closure_eq
IsClosed.preimage
isClosed_closure
closure_mono
Set.preimage_mono
subset_closure
comp 📖Continuouscontinuous_def
IsOpen.preimage
comp' 📖Continuouscomp
continuousAt 📖mathematicalContinuousContinuousAttendsto
frontier_preimage_subset 📖mathematicalContinuousSet
Set.instHasSubset
frontier
Set.preimage
Set.diff_subset_diff
closure_preimage_subset
preimage_interior_subset_interior_preimage
iterate 📖mathematicalContinuousNat.iteratecontinuous_id
comp
range_subset_closure_image_dense 📖mathematicalContinuous
Dense
Set
Set.instHasSubset
Set.range
closure
Set.image
Set.image_univ
Dense.closure_eq
image_closure_subset_closure_image
tendsto 📖mathematicalContinuousFilter.Tendsto
nhds
Filter.HasBasis.tendsto_iff
nhds_basis_opens
IsOpen.preimage
Set.Subset.rfl
tendsto' 📖mathematicalContinuousFilter.Tendsto
nhds
tendsto

ContinuousAt

Theorems

NameKindAssumesProvesValidatesDepends On
comp 📖ContinuousAtFilter.Tendsto.comp
comp' 📖ContinuousAtcomp
comp_of_eq 📖ContinuousAtcomp
eventually_mem 📖mathematicalContinuousAt
Set
Filter
Filter.instMembership
nhds
Filter.Eventually
Set.instMembership
iterate 📖mathematicalContinuousAtNat.iteratecontinuousAt_id
comp_of_eq
preimage_mem_nhds 📖mathematicalContinuousAt
Set
Filter
Filter.instMembership
nhds
Set.preimage
tendsto 📖mathematicalContinuousAtFilter.Tendsto
nhds

Dense

Theorems

NameKindAssumesProvesValidatesDepends On
denseRange_val 📖mathematicalDenseDenseRange
Set
Set.instMembership
denseRange_subtype_val

DenseRange

Definitions

NameCategoryTheorems
some 📖CompOp

Theorems

NameKindAssumesProvesValidatesDepends On
closure_range 📖mathematicalDenseRangeclosure
Set.range
Set.univ
Dense.closure_eq
comp 📖DenseRange
Continuous
eq_1
Set.range_comp
dense_image
dense_image 📖mathematicalDenseRange
Continuous
Dense
Set.imageDense.of_closure
Dense.mono
Continuous.range_subset_closure_image_dense
dense_of_mapsTo 📖DenseRange
Continuous
Dense
Set.MapsTo
Dense.mono
Set.MapsTo.image_subset
dense_image
exists_mem_open 📖mathematicalDenseRange
IsOpen
Set.Nonempty
Set
Set.instMembership
Set.exists_range_iff
Dense.exists_mem_open
mem_nhds 📖mathematicalDenseRange
Set
Filter
Filter.instMembership
nhds
Set.instMembershipexists_mem_open
isOpen_interior
mem_interior_iff_mem_nhds
interior_subset
nonempty 📖DenseRangenonempty_iff
nonempty_iff 📖DenseRangeSet.range_nonempty_iff_nonempty
Dense.nonempty_iff
subset_closure_image_preimage_of_isOpen 📖mathematicalDenseRange
IsOpen
Set
Set.instHasSubset
closure
Set.image
Set.preimage
Set.image_preimage_eq_inter_range
Dense.open_subset_closure_inter

Equiv

Theorems

NameKindAssumesProvesValidatesDepends On
continuous_symm_iff 📖mathematicalContinuous
DFunLike.coe
Equiv
EquivLike.toFunLike
instEquivLike
symm
IsOpenMap
isOpenMap_symm_iff 📖mathematicalIsOpenMap
DFunLike.coe
Equiv
EquivLike.toFunLike
instEquivLike
symm
Continuous

Filter.EventuallyEq

Theorems

NameKindAssumesProvesValidatesDepends On
continuousAt 📖mathematicalFilter.EventuallyEq
nhds
ContinuousAtcontinuousAt_congr
tendsto_const_nhds

Filter.Tendsto

Theorems

NameKindAssumesProvesValidatesDepends On
lift'_closure 📖mathematicalContinuous
Filter.Tendsto
Filter.lift'
closure
Filter.tendsto_lift'
Filter.mp_mem
Filter.mem_lift'
Filter.univ_mem'
Set.MapsTo.closure
Set.mapsTo_preimage

Function.Surjective

Theorems

NameKindAssumesProvesValidatesDepends On
denseRange 📖mathematicalDenseRangerange_eq
IsClosed.closure_eq

IsClosed

Theorems

NameKindAssumesProvesValidatesDepends On
preimage 📖mathematicalContinuousIsClosed
Set.preimage
continuous_iff_isClosed

IsOpen

Theorems

NameKindAssumesProvesValidatesDepends On
preimage 📖mathematicalContinuous
IsOpen
Set.preimageContinuous.isOpen_preimage

LibraryNote

Definitions

NameCategoryTheorems
comp_of_eq_lemmas 📖CompOp
continuity_lemma_statement 📖CompOp

Set.MapsTo

Theorems

NameKindAssumesProvesValidatesDepends On
closure 📖mathematicalSet.MapsTo
Continuous
closureClusterPt.map
Continuous.continuousAt
Filter.tendsto_principal_principal
closure_left 📖mathematicalSet.MapsTo
Continuous
closureclosure
IsClosed.closure_eq

(root)

Definitions

NameCategoryTheorems
Continuous 📖CompData
966 mathmath: continuous_decomposeProdAdjoint, Valued.continuous_extension, isProperMap_iff_tendsto_cocompact, ModelWithCorners.continuous_invFun, NormedSpace.exp_continuous, continuous_mabs, Equicontinuous.continuous, completelyRegularSpace_iff_isOpen, ContinuousLinearMap.continuous₂, ContinuousMap.continuous, ENNReal.continuous_nnreal_sub, IsSpectralMap.continuous, PowerSeries.WithPiTopology.continuous_coeff, continuous_zpow, continuous_coinduced_dom, OrderIso.continuous, ContinuousLinearMap.continuous_det, Polynomial.continuous_aeval, OnePoint.continuous_iff, Matrix.SpecialLinearGroup.continuous_toGL, continuous_nsmul, AddCircle.liftIoc_zero_continuous, IsOpenQuotientMap.continuous, ContinuousMap.exists_finite_approximation_of_mem_nhds_diagonal, continuous_zsmul, AlternatingMap.continuous_of_bound, continuous_sup_dom, ContinuousLinearMap.continuous_integral_comp_L1, ContinuousLineDeriv.continuous_lineDerivOp, isProperMap_iff_ultrafilter, Topology.WithUpper.continuous_toUpper, SeparationQuotient.continuous_mk, HasFTaylorSeriesUpTo.cont, Real.binEntropy_continuous, Metric.PiNatEmbed.continuous_distDenseSeq_inv, Rat.continuous_coe_real, ContinuousMap.continuous_prodMk_const, RestrictedProduct.continuous_rng_of_principal, mdifferentiable_iff_target, IsBoundedBilinearMap.continuous_left, continuous_sSup_dom, Quaternion.continuous_imI, continuous_sumMap, isOpen_iff_continuous_mem, MeasureTheory.Integrable.continuous_primitive, PiNat.exists_retraction_subtype_of_isClosed, IsProperMap.toContinuous, continuous_of_tsupport, unitInterval.toNNReal_continuous, Topology.IsClosedEmbedding.isClosedEmbedding_iff_continuous_injective_isClosedMap, Real.continuous_rpow_const, LightCondSet.continuous_coinducingCoprod, Pullback.continuous_lift, Real.continuous_mul_log, Polynomial.continuous, TopCommRingCat.instReflectsIsomorphismsTopCatForget₂SubtypeRingHomαContinuousCoeContinuousMapCarrier, ContinuousConstSMul.continuous_const_smul, CofiniteTopology.continuous_of, Real.continuous_arcsin, continuous_stoneCechUnit, AddCircle.liftIco_continuous, RestrictedProduct.continuous_coe, Seminorm.continuous_of_forall', ContDiff.continuous_deriv_one, intervalIntegral.continuous_of_dominated_interval, ContinuousVAdd.continuous_vadd, LinearMap.continuous_of_seq_closed_graph, List.Vector.continuous_insertIdx', Complex.continuous_sin, ContinuousSemilinearEquivClass.map_continuous, Seminorm.continuous', Topology.IsUpperSet.monotone_iff_continuous, Topology.IsUpper.continuous_iff_Iic, MeasureTheory.Integrable.exists_hasCompactSupport_lintegral_sub_le, AffineEquiv.continuous_of_finiteDimensional, MeasureTheory.analyticSet_iff_exists_polishSpace_range, ContinuousAlgEquiv.continuous_invFun, ContinuousAddMonoidHom.continuous_comp, BoundedContinuousFunction.continuous_compContinuous, continuous_equivFun_basis, contDiff_one_iff_fderiv, continuous_extendFrom, continuous_sumElim, Real.continuous_sinc, MvPowerSeries.continuous_aeval, ContinuousStar.continuous_star, TestFunction.continuous_iff_continuous_comp, continuous_of_linear_of_boundₛₗ, contDiff_zero, RCLike.continuous_normSq, isProperMap_iff_clusterPt, continuous_const_cpow, cfcHom_continuous, Pi.continuous_precomp, DiscreteQuotient.ofLE_continuous, IsLocalHomeomorph.continuous, ContMDiff.continuous, VectorBundleCore.continuous_proj, TrivSqZeroExt.continuous_snd, continuous_inl, AffineIsometryEquiv.comp_continuous_iff, continuous_prod_of_discrete_right, Int.cast_continuous, ContDiff.continuous_deriv, continuous_congr_of_inseparable, AffineMap.homothety_continuous, NormedSpace.Dual.toWeakDual_continuous, isProperMap_iff_ultrafilter_of_t2, MeasureTheory.ProbabilityMeasure.continuous_pi, continuous_subtype_val, Isometry.comp_continuous_iff, MeasureTheory.Measure.AddMonoidHom.continuous_of_measurable, DomMulAct.continuous_mk, IsTopologicalAddGroup.continuous_conj', Path.Homotopy.continuous_transAssocReparamAux, continuous_sigma_map, continuous_of_locally_uniform_approx_of_continuousAt, AddMonoidHom.continuous_of_isBounded_nhds_zero, Homeomorph.continuous, FiberPrebundle.continuous_proj, MeasureTheory.ProbabilityMeasure.continuous_lintegral_boundedContinuousFunction, ContinuousNeg.continuous_neg, SeparationQuotient.continuous_lift, continuous_if', ContinuousMap.continuous_postcomp, MeasureTheory.continuous_condExpIndL1, GroupTopology.continuous_mul', continuous_of_linear_of_bound, ContinuousMap.continuous_iff_continuous_uniformOnFun, SmoothBumpFunction.continuous, UniformEquiv.continuous_symm, ENNReal.continuous_log, MeasureTheory.LevyProkhorov.continuous_equiv_symm_probabilityMeasure, AddMonoidHomClass.continuous_of_bound, Flow.cont', SeqContinuous.continuous, WithLp.continuous_fst, MeasureTheory.FiniteMeasure.continuous_testAgainstNN_eval, HomeomorphClass.inv_continuous, LocallyConstant.continuous, Topology.IsLower.continuous_iff_Ici, ContinuousSemilinearEquivClass.inv_continuous, MonoidHomClass.continuous_of_bound, ContinuousOn.restrict_mapsTo, Set.Equicontinuous.continuous_of_mem, MeasureTheory.L1.continuous_integral, TopologicalSpace.Closeds.continuous_closure, ContinuousAlgEquiv.continuous, IsBoundedLinearMap.continuous, SemilinearIsometryClass.continuous, ExistsContDiffBumpBase.u_continuous, Measurable.exists_continuous, SimplexCategory.continuous_toTopMap, PowerSeries.WithPiTopology.continuous_constantCoeff, PiLp.continuous_toLp, ContinuousMap.continuous_compactOpen, Topology.IsInducing.continuous_iff, Real.continuous_tan, MeasureTheory.continuous_diracProbaEquiv, continuous_of_const, continuous_quotient_mk', CommRingCat.HomTopology.continuous_precomp, continuous_mul, LaurentSeries.continuous_coe, Topology.IsCoherentWith.continuous_iff, Path.continuous_symm, exists_retractionCantorSet, MeasureTheory.continuous_setToFun, Units.continuous_val, continuous_iff_lower_upperSemicontinuous, TestFunction.continuous_ofSupportedIn, ModularFormClass.continuous, Metric.continuous_iff, Path.continuous_extend, isProperMap_iff_universally_closed, continuous_toAdd, Complex.continuous_re, ENNReal.continuous_sub_right, PartitionOfUnity.IsSubordinate.continuous_finsum_smul, hasFTaylorSeriesUpTo_zero_iff, Valued.continuous_valuation, continuous_of_continuousAt_zero₂, FormalMultilinearSeries.continuousOn_of_finite, Real.continuous_log, continuous_const, MeasureTheory.LevyProkhorov.continuous_equiv_probabilityMeasure, SchwartzMap.continuous, continuous_id_of_le, ContinuousAddEquiv.continuous_toFun, ContinuousMap.continuous_toNNReal, Real.continuous_sqrt, Bundle.ContinuousRiemannianMetric.continuous, IsModuleTopology.continuous_bilinear_of_pi_fintype, AbstractCompletion.continuous_coe, continuous_algebraMap, ContDiff.continuous_fderiv_apply, Matrix.GeneralLinearGroup.continuous_det, continuous_of_continuousAt_zero, List.continuous_sum, Pi.continuous_precomp', NNReal.continuous_coe, Complex.continuous_cosh, continuous_inr, TopologicalSpace.NonemptyCompacts.continuous_toCompacts, FunOnFinite.continuous_linearMap, DiscreteQuotient.proj_continuous, continuous_iInf_rng, PadicInt.continuous_choose, ContDiff.continuous, MvPowerSeries.WithPiTopology.continuous_coeff, continuous_dist, Homeomorph.continuous_toFun, Order.IsNormal.continuous, ENNReal.continuous_const_mul, continuous_rangeFactorization_iff, exists_nat_nat_continuous_surjective_of_completeSpace, TopCat.continuousPrelocal_pred, continuous_decomposeProdAdjoint_symm, continuous_uliftUp, AffineIsometry.comp_continuous_iff, continuous_div_right', Flow.IsSemiconjugacy.cont, continuous_id', WithSeminorms.uniformEquicontinuous_iff_exists_continuous_seminorm, continuous_bot, DomAddAct.continuous_mk_symm, FormalMultilinearSeries.partialSum_continuous, continuous_sigmaMk, AffineMap.lineMap_continuous, ContinuousLinearMap.continuous_restrictScalars, Path.continuous_delayReflRight, Real.qaryEntropy_continuous, RingTopology.coinduced_continuous, AffineMap.continuous_iff, continuous_to_deltaGenerated, LightProfinite.continuous_iff_convergent, Path.continuous_delayReflLeft, continuous_apply_apply, Topology.isGeneratedBy_iff, ContinuousSup.continuous_sup, ContinuousMultilinearMap.range_toUniformOnFun, Continuous.prodMk_right, contMDiff_zero_iff, TopCommRingCat.forgetToTopCatTopologicalRing, Complex.continuous_circleTransform, MeasureTheory.Measure.innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_group, RestrictedProduct.continuous_dom, Seminorm.continuous, sInfHom.continuous, Real.continuous_cosh, ContDiff.continuous_fderiv, continuous_polarCoord_symm, continuous_of_le_add_edist, continuous_ofDual, TopologicalSpace.Compacts.continuous_coe, IsLocallyConstant.iff_continuous, Dilation.toContinuous, RestrictedProduct.continuous_dom_prod, Topology.IsLowerSet.monotone_iff_continuous, Nat.cast_continuous, CategoryTheory.PreGaloisCategory.continuous_mapAut_whiskeringRight, Quaternion.continuous_imJ, Topology.IsGeneratedBy.continuous_equiv_symm, ContDiffMapSupportedIn.continuous_iff_comp_withOrder, AlgebraicGeometry.LocallyRingedSpace.toΓSpec_continuous, ContDiff.continuous_zero, ContinuousAddMonoidHom.continuous_comp_right, ContinuousMulEquiv.continuous_toFun, isProperMap_iff_isCompact_preimage, continuous_edist, continuous_iff_preimage_interior_subset_interior_preimage, ContinuousENorm.continuous_enorm, RestrictedProduct.continuous_rng_of_top, CFC.continuous_abs, contDiff_iff_iteratedDeriv, continuous_sum_swap, Real.continuous_probChar, continuous_bool_rng, AnalyticOnNhd.continuous, DomMulAct.continuous_mk_symm, continuous_add_right, WithTop.continuous_coe, continuous_snd, continuous_fst, IsValuativeTopology.continuous_valuation, equicontinuous_finite, InfiniteGalois.mulEquivToLimit_symm_continuous, selfAdjoint.continuous_expUnitary, ENNReal.continuous_coe_iff, MeasureTheory.ProbabilityMeasure.continuous_iff_forall_continuousMap_continuous_integral, AddUnits.continuous_val, Finset.continuous_restrict₂, unitInterval.continuous_sigmoid, Complex.continuous_sinh, AffineIsometryEquiv.continuous, continuous_isRight, LinearMap.IsSymmetric.continuous, MaximalSpectrum.toPrimeSpectrum_continuous, Preorder.continuous_frestrictLe, CPolynomialOn.continuous, PrimeSpectrum.continuous_comap, Topology.IsOpenEmbedding.continuous, List.Vector.continuous_eraseIdx, FiberBundle.continuous_totalSpaceMk, Ultrafilter.continuous_add_left, Profinite.NobelingProof.continuous_swapTrue, Homeomorph.comp_continuous_iff', continuous_sup, contDiff_iff_continuous_differentiable, ContinuousOn.continuous_of_mulTSupport_subset, FiberBundleCore.continuous_const_section, CircleDeg1Lift.continuous_iff_surjective, ContinuousAlgEquiv.continuous_toFun, continuous_mul_right, AddCircle.continuous_equivIoc_symm, stdSimplex.continuous_map, MeasureTheory.FiniteMeasure.continuous_iff_forall_continuous_lintegral, continuous_gaugeRescale, continuous_top, ContinuousAlgEquiv.comp_continuous_iff, Valued.continuous_extensionValuation, mulRight_continuous, contDiff_nat_iff_iteratedDeriv, AddOpposite.continuous_op, LinearMap.continuous_of_isContPerfPair, periodizedBernoulli.continuous, GenLoop.continuous_fromLoop, contMDiff_iff_target, IsBoundedBilinearMap.continuous, continuous_min, continuous_swap, Complex.continuous_circleTransformDeriv, ContDiff.continuous_iteratedDeriv, Diffeology.DSmooth.continuous, MeasureTheory.Integrable.exists_hasCompactSupport_integral_sub_le, Complex.continuous_conj, Real.continuous_sin, ContinuousMulEquiv.continuous_invFun, NumberField.mixedEmbedding.fundamentalCone.continuous_expMap, continuous_circleMap, ContinuousMultilinearMap.coe_continuous, ContinuousAddEquiv.continuous_invFun, MeasureTheory.FiniteMeasure.continuous_iff_forall_continuousMap_continuous_lintegral, Real.continuous_negMulLog, UniformOnFun.continuous_rng_iff, WithSeminorms.uniformEquicontinuous_iff_bddAbove_and_continuous_iSup, WeakDual.exists_countable_separating, mdifferentiable_iff, Function.Periodic.continuous_qParam, Matrix.GeneralLinearGroup.continuous_upperRightHom, Real.continuous_const_rpow, OpenPartialHomeomorph.continuous_univBall, ENNReal.continuous_ofReal, Monotone.continuous_of_surjective, continuous_if, ENNReal.continuous_pow, Complex.continuous_exp, continuous_gauge, BoundedContinuousFunction.continuous, isHomeomorph_iff_continuous_bijective, ContinuousOn.comp_fract', LinearMap.mem_span_iff_continuous_of_finite, IsContinuousRiemannianBundle.exists_continuous, IsOpenQuotientMap.continuous_comp_iff, MulOpposite.continuous_unop, Profinite.NobelingProof.continuous_projRestrict, ENNReal.continuous_zpow, MvPowerSeries.continuous_subst, UniformSpace.Completion.continuous_coeRingHom, GromovHausdorff.toGHSpace_continuous, LinearIsometry.continuous, t1Space_TFAE, continuous_bernoulliFun, Order.isNormal_iff_strictMono_and_continuous, Profinite.NobelingProof.continuous_CC'₁, Finset.continuous_restrict, DeltaGeneratedSpace.continuous_counit, Metric.continuous_iff', AddOpposite.continuous_unop, Homeomorph.continuous_symm, NumberField.mixedEmbedding.convexBodySumFun_continuous, exists_nat_bool_continuous_surjective_of_compact, RCLike.continuous_re, ContinuousConstVAdd.continuous_const_vadd, IsTopologicalGroup.continuous_conj', isProperMap_iff_isClosedMap_and_tendsto_cofinite, Homeomorph.comp_continuous_iff, MeasureTheory.continuous_integral_integral, ChartedSpace.liftProp_iff, Dense.continuous_extend, Real.continuous_logb, Continuous.homeoOfEquivCompactToT2.t1_counterexample, continuous_one, BumpCovering.continuous_toPOUFun, continuous_iSup_dom, Homeomorph.continuous_sumAssoc, ContinuousDiv.continuous_div', continuous_coe_ennreal_ereal, continuous_enorm, isProperMap_iff_isClosedMap_ultrafilter, Real.continuous_logb', IsTopologicalAddGroup.continuous_conj, ContinuousFourierInv.continuous_fourierInv, ContinuousMap.continuous_const', ProbabilityTheory.Kernel.continuous_integral_integral, unitInterval.continuous_qRight, IsModuleTopology.continuous_bilinear_of_finite_left, equicontinuous_iff_continuous, AddCircle.continuous_toCircle, ContinuousMul.continuous_mul, ContinuousMap.continuous_iff_continuous_uniformFun, MeasureTheory.MemLp.exists_hasCompactSupport_integral_rpow_sub_le, continuous_iff_le_induced, Topology.continuous_reorderRestrictProd, ContinuousAlgHom.continuous, NNReal.continuous_sqrt, continuous_inf, Path.Homotopy.continuous_reflTransSymmAux, continuous_coe_real_ereal, sSupHom.continuous, BoundedContinuousFunction.continuous_coe, Real.continuous_sinh, continuous_IccExtend_iff, ContinuousMapZero.continuous_comp_left, continuous_iff_isClosed, Topology.IsGeneratedBy.equiv_symm_comp_continuous_iff, Complex.continuous_normSq, List.continuous_insertIdx, MeasureTheory.ProbabilityMeasure.continuous_lintegral_continuousMap, TopologicalSpace.Closeds.continuous_infEDist, NumberField.mixedEmbedding.fundamentalCone.continuous_expMapBasis, mulLeft_continuous, OpenPartialHomeomorph.continuous_iff_continuous_comp_left, Path.continuous_uncurry_iff, WithLp.prod_continuous_toLp, continuous_div_left', continuous_zero, continuous_clm_apply, RCLike.continuous_ofReal, Pullback.continuous_proj, continuous_mulIndicator, Real.continuous_log', ContinuousLinearMap.range_coeFn_eq, MeasureTheory.ProbabilityMeasure.continuous_integral_boundedContinuousFunction, PowerSeries.WithPiTopology.continuous_C, IsOpen.measure_eq_biSup_integral_continuous, ContinuousOrderHom.continuous_toFun, IsModuleTopology.continuous_of_linearMap, PartitionOfUnity.continuous_finsum_smul, NumberField.mixedEmbedding.continuous_norm, IsMIntegralCurve.continuous, LinearMap.continuous_of_locally_bounded, Homeomorph.continuous_sumAssoc_symm, MultilinearMap.continuous_of_bound, AddGroupTopology.continuous_add', Topology.IsClosedEmbedding.continuous, PadicInt.continuous_addChar_of_value_at_one, UniformContinuous.continuous, PiLp.continuous_apply, ContinuousMonoidHom.continuous_comp_left, ContinuousMonoidHom.continuous_toFun, CondensedSet.continuous_coinducingCoprod, QuotientGroup.continuous_mk, MeasureTheory.MemLp.exists_hasCompactSupport_eLpNorm_sub_le, continuous_inner, ENNReal.continuous_mul_const, Seminorm.continuous_iff, IsHomeomorph.continuous, continuous_of_continuousOn_iUnion_of_isOpen, WithSeminorms.continuous_seminorm, Isometry.continuous, PadicInt.continuousAddCharEquiv_of_norm_mul_symm_apply, WeakDual.coeFn_continuous, ConnectedComponents.continuous_coe, ContinuousAlternatingMap.continuous_restrictScalars, continuous_add_left, UniformSpace.Completion.continuous_coe, Perfect.exists_nat_bool_injection, ContDiff.continuous_iteratedFDeriv', continuous_pi_iff, ContinuousMap.continuous_coev, Units.continuous_iff, ContinuousLinearMap.continuous_uncurry_of_multilinear, continuous_norm', ContinuousMap.continuous_restrict, RestrictedProduct.continuous_inclusion, Real.continuous_exp, TopologicalSpace.Closeds.continuous_singleton, TrivSqZeroExt.continuous_inl, LinearMap.continuous_of_nonzero_on_open, gronwallBound_continuous_ε, continuous_prod_of_discrete_left, ModelWithCorners.continuous_toFun, ContinuousSub.continuous_sub, Complex.continuous_cos, Pullback.continuous_totalSpaceMk, ContinuousMap.continuous_uncurry, Real.continuous_arctan, DomAddAct.continuous_mk, Metric.PiNatEmbed.continuous_distDenseSeq, LipschitzWith.continuous, MeasureTheory.ProbabilityMeasure.continuous_iff_forall_continuous_lintegral, ContinuousLinearMap.continuous, Diffeology.DSmooth.continuous', LinearIsometry.comp_continuous_iff, continuous_matrix_diag, Trivialization.continuous_coordChange, equicontinuous_unique, ContinuousOn.continuous_restrict_iff_continuous_uniformOnFun, CompletelyRegularSpace.completely_regular, Metric.Snowflaking.continuous_toSnowflaking, Module.Basis.continuous_toMatrix, OrderTopology.continuous_iff, RCLike.continuous_conj, continuous_congr, FiberPrebundle.continuous_totalSpaceMk, TopCat.continuous_iff_of_isColimit, continuous_barycentric_coord, ENNReal.continuous_sub_left, CommRingCat.HomTopology.continuous_apply, ContinuousMap.continuous_isUnit_unit, Seminorm.cont_withSeminorms_normedSpace, Topology.WithGeneratedByTopology.continuous_from_iff, continuous_id, continuous_update, ContinuousMultilinearMap.cont, IsDenseInducing.continuous, Real.continuous_inv, Path.continuous, Compactum.continuous_of_hom, continuous_toDual, continuous_id_iff_le, Pi.continuous_restrict, T2Quotient.continuous_mk, ContinuousLinearEquiv.continuous_invFun, ContDiffBump.continuous, continuous_inclusion, ModelWithCorners.continuous, Topology.isOpenEmbedding_iff_continuous_injective_isOpenMap, ENNReal.continuous_exp, RestrictedProduct.continuous_rng_of_principal_iff_forall, Dilation.comp_continuous_iff, IsDenseInducing.continuous_extend, ProbabilityTheory.Kernel.continuous_integral_integral_comp, UniformEquiv.continuous, Real.continuous_cos, ContinuousAlternatingMap.range_toAlternatingMap, ModelWithCorners.continuous_symm, LocallyLipschitz.continuous, Quaternion.continuous_im, continuous_boolIndicator_iff_isClopen, continuous_sigma_iff, WeakBilin.coeFn_continuous, Real.Angle.continuous_coe, isOpenQuotientMap_iff, contMDiff_iff, LinearMap.continuous_of_finiteDimensional, IsDenseInducing.continuous_extend_of_cauchy, Seminorm.continuous_from_bounded, IsTopologicalAddTorsor.continuous_vsub, continuous_indicator, continuous_selfAdjointPart, ContinuousAffineEquiv.continuous, Path.subpathAux_continuous, continuous_quot_mk, NumberField.mixedEmbedding.continuous_normAtPlace, UpperHalfPlane.continuous_im, IsometryEquiv.comp_continuous_iff', PiLp.continuous_ofLp, HomeomorphClass.map_continuous, LineDeriv.continuous_iteratedLineDerivOp, MeasureTheory.ProbabilityMeasure.continuous_prod, MeasureTheory.continuous_integral, Topology.IsQuotientMap.continuous, PolishSpace.exists_nat_nat_continuous_surjective, SchwartzMap.convolution_continuous_left, ContDiff.continuous_iteratedFDeriv, continuous_negPart, ContinuousMonoidHom.continuous_comp_right, expNegInvGlue.continuous_polynomial_eval_inv_mul, ContinuousEvalConst.continuous_eval_const, ContinuousMap.continuous_uncurry_of_continuous, continuous_ofMul, Diffeology.IsPlot.continuous, continuous_map_sInf, Complex.UnitDisc.continuous_coe, ContinuousMap.continuous_toFun, SchwartzMap.pairing_continuous_left, Homeomorph.continuous_invFun, AddCircle.continuous_mk', PolynormableSpace.withSeminorms', t1Space_iff_continuous_cofinite_of, continuous_algebraMap_iff_smul, EReal.continuous_coe_iff, EReal.continuous_toENNReal, continuousOn_iff_continuous_restrict, IsClosed.exists_nat_bool_injection_of_not_countable, continuous_uliftDown, RestrictedProduct.continuous_dom_pi, IsBoundedBilinearMap.continuous_right, contDiff_nat_iff_continuous_differentiable, OnePoint.continuous_iff_from_discrete, Diffeomorph.continuous, MDifferentiable.continuous, ContinuousLinearEquiv.continuous, SingularManifold.hf, LinearMap.continuous_on_pi, IsModuleTopology.continuous_neg, TrivSqZeroExt.continuous_inr, TopologicalSpace.Compacts.continuous_toCloseds, ODE.FunSpace.continuous_compProj, AddGroupTopology.coinduced_continuous, Topology.IsScott.ωScottContinuous_iff_continuous, LocallyFinite.continuous', upperHemicontinuous_singleton_iff, ContinuousMap.Homotopy.continuous, Profinite.NobelingProof.continuous_projRestricts, HasFTaylorSeriesUpTo.continuous, AddCircle.liftIoc_continuous, Complex.continuous_ofReal_cpow_const, FunOnFinite.continuous_map, Path.continuous_trans, ContinuousOn.restrict, SchwartzMap.continuous_toLp, separatesPoints_continuous_of_t35Space_Icc, isProperMap_iff_isClosedMap_filter, continuous_single, UniformSpace.Completion.continuous_map, IsHomeomorphicTrivialFiberBundle.continuous_proj, contDiff_one_iff_hasFDerivAt, NNReal.continuous_rpow_const, UniformSpace.Completion.continuous_extension, MeasureTheory.continuous_indicatorConstLp_set, NNReal.continuous_nnrpow_const, Quaternion.continuous_coe, Complex.continuous_ofReal, InformationTheory.continuous_klFun, Metric.continuous_infNndist_pt, continuous_inv_iff, continuous_ultrafilter_extend, UniformSpace.hausdorff.continuous_closure, continuous_induced_dom, Uniform.continuous_iff'_right, TopologicalSpace.IsTopologicalBasis.continuous_iff, continuous_add, AddCircle.continuous_equivIco_symm, ENNReal.continuous_truncateToReal, Path.subpath_continuous_family, ContinuousMap.continuous_mkD_restrict_of_uncurry, continuous_real_toNNReal, Units.continuous_coe_inv, MeasureTheory.MeasuredSets.continuous_measure, Real.smoothTransition.continuous, PadicInt.continuous_multichoose, Path.Homotopy.continuous_transReflReparamAux, cantorToHilbert_continuous, Metric.continuous_infDist_pt, continuous_projIcc, Real.Angle.continuous_sin, continuous_piecewise, IsProperMap.continuous, continuous_of_continuousAt_one, ContinuousLinearMapWOT.continuous_inducingFn, OnePoint.not_continuous_cofiniteTopology_of_symm, MeasureTheory.FiniteMeasure.continuous_lintegral_continuousMap, MeasureTheory.ProbabilityMeasure.continuous_iff_forall_continuous_integral, AffineMap.continuous_linear_iff, NumberField.mixedEmbedding.continuous_normAtAllPlaces, InfiniteGalois.restrictNormalHom_continuous, ContinuousLinearMap.cont, continuous_prodMk, PowerSeries.continuous_aeval, MeasureTheory.Lp.continuous_negPart, Equiv.continuous_symm_iff, ContinuousFourier.continuous_fourier, BoundedContinuousFunction.continuous_eval_const, AbstractCompletion.continuous_map, IsCoveringMap.eq_liftHomotopy_iff, continuous_map_of_le, BoundedContinuousFunction.continuous_eval, cfcₙHom_continuous, ODE.FunSpace.continuous, ContinuousMap.continuous_comp', MeasureTheory.ProbabilityMeasure.toWeakDualBCNN_continuous, FiberBundle.continuous_proj, cfcₙHomSuperset_continuous, continuous_nnnorm, MeasureTheory.FiniteMeasure.continuous_integral_boundedContinuousFunction, ContinuousAffineEquiv.continuous_toFun, ContinuousLinearEquiv.comp_continuous_iff, Real.continuous_fourierChar, AffineMap.lineMap_continuous_uncurry, RestrictedProduct.continuous_dom_prod_right, Filter.Tendsto.continuous_of_equicontinuous, continuous_const_vadd_iff, continuous_apply, ContinuousAffineEquiv.continuous_invFun, Metric.continuous_infEDist, UniformOnFun.isClosed_setOf_continuous, TopologicalSpace.NonemptyCompacts.continuous_toCloseds, Seminorm.continuous_of_lowerSemicontinuous, continuous_of_cover_nhds, Units.continuous_embedProduct, Path.truncate_const_continuous_family, OnePoint.continuous_map_iff, MeasureTheory.continuous_diracProba, WeakBilin.eval_continuous, MeasureTheory.LevyProkhorov.continuous_ofMeasure_probabilityMeasure, LinearMap.continuous_of_isClosed_ker, Topology.WithGeneratedByTopology.continuous_equiv, continuous_Prop, QuotientAddGroup.continuous_mk, GroupTopology.continuous_inv', AlgebraicGeometry.Scheme.Hom.continuous, IsModuleTopology.continuous_of_distribMulActionHom, IsCompactOperator.continuous, IsCompact.measure_eq_biInf_integral_hasCompactSupport, IsTopologicalGroup.continuous_conj_prod, LinearMap.continuous_iff_isClosed_ker, MeasureTheory.FiniteMeasure.continuous_mass, PrimeSpectrum.continuous_tensorProductTo, isProperMap_iff_isClosedMap_and_compact_fibers, ContinuousSMul.continuous_smul, OnePoint.continuous_coe, MeasureTheory.FiniteMeasure.toWeakDualBCNN_continuous, separatesPoints_continuous_of_t35Space, Complex.continuous_tan, LinearMap.continuous_of_isClosed_graph, Path.truncate_continuous_family, ContinuousEval.continuous_eval, ContinuousOn.mapsToRestrict, CStarModule.continuous_inner, UniformSpace.Completion.continuous_toCompl, RestrictedProduct.continuous_eval, Topology.WithLower.continuous_toLower, ContinuousInf.continuous_inf, MeasureTheory.ProbabilityMeasure.continuous_testAgainstNN_eval, ContinuousLinearMap.reApplyInnerSelf_continuous, continuous_thickenedIndicatorAux, Profinite.NobelingProof.continuous_CC'₀, List.continuous_cons, TrivSqZeroExt.continuous_fst, ContinuousMap.continuous_precomp, MeasureTheory.AnalyticSet_def, isHomeomorph_iff_exists_inverse, Pi.continuous_restrict₂, continuous_const_smul_iff₀, ContinuousMap.HomotopyWith.continuous, Real.continuous_ofDigits, MeasureTheory.continuous_diracProbaEquivSymm, PadicInt.continuousAddCharEquiv_symm_apply, continuous_descPochhammer_eval, Topology.IsQuotientMap.continuous_iff, LipschitzWith.continuous_compLp, Topology.IsUpperSet.monotone_to_upperTopology_continuous, isHomeomorph_iff_continuous_isClosedMap_bijective, ContinuousMapClass.map_continuous, ContDiffMapSupportedIn.continuous_iff_comp, ContinuousAlgEquiv.comp_continuous_iff', ProfiniteGrp.toLimitFun_continuous, EMetric.NonemptyCompacts.continuous_toCloseds, Action.isContinuous_def, ContinuousLinearMap.toUniformConvergenceCLM_continuous, AddCircle.liftIco_zero_continuous, MeasureTheory.ProbabilityMeasure.continuous_integral_continuousMap, cfcHomSuperset_continuous, TopologicalSpace.NonemptyCompacts.continuous_coe, continuous_mulSingle, continuous_iff_coinduced_le, IsometryClass.continuous, ContinuousMap.compRightAlgHom_continuous, MeasureTheory.Lp.compMeasurePreserving_continuous, MeasureTheory.ProbabilityMeasure.toFiniteMeasure_continuous, MeasureTheory.FiniteMeasure.continuous_iff_forall_continuousMap_continuous_integral, ENNReal.continuous_coe, InfiniteGalois.algEquivToLimit_continuous, ENNReal.continuous_rpow_const, continuous_nnnorm', HolderWith.continuous, Urysohns.CU.continuous_lim, IsModuleTopology.continuous_bilinear_of_finite_right, FiberBundleCore.continuous_proj, MeasureTheory.LevyProkhorov.continuous_toMeasure_probabilityMeasure, completelyRegularSpace_iff, IsCoveringMap.continuous, EReal.continuous_coe_ennreal_iff, RCLike.continuous_im, Profinite.NobelingProof.continuous_proj, continuousOn_univ, continuous_neg_iff, continuous_const_smul_iff, LinearIsometryEquiv.comp_continuous_iff, continuous_coeFun, Fin.continuous_append, MeasureTheory.ProbabilityMeasure.continuous_iff_forall_continuousMap_continuous_lintegral, StarAlgebra.elemental.continuous_characterSpaceToSpectrum, MeasureTheory.FiniteMeasure.continuous_integral_continuousMap, LinearPMap.mem_adjoint_domain_iff, SeparationQuotient.continuous_lift₂, UniformSpace.Completion.continuous_inner, ContinuousMap.HomotopyEquiv.continuous, ContinuousAlgHom.cont, AkraBazziRecurrence.continuous_sumCoeffsExp, Equiv.isOpenMap_symm_iff, Topology.IsInducing.continuous, continuous_of_discreteTopology, Seminorm.continuous_of_continuousAt_zero, Module.Basis.continuous_coe_repr, UniformOnFun.continuous_of_forall_lipschitzWith, IsLocallyConstant.continuous, MeasureTheory.FiniteMeasure.continuous_iff_forall_continuous_integral, ProbabilityTheory.continuous_mgf, continuous_isLeft, ContinuousLinearEquiv.continuous_toFun, continuous_inf_rng, Topology.IsEmbedding.continuous_iff, continuous_posPart, Flow.continuous_toFun, PiNat.exists_retraction_of_isClosed, PolynormableSpace.withSeminorms, Filter.continuous_nhds, Monotone.continuous_of_denseRange, LinearMap.IsContPerfPair.continuous_uncurry, ContinuousInv.continuous_inv, ContinuousOn.continuous_of_tsupport_subset, continuous_abs, Continuous.prodMk_left, ContinuousMonoidHom.continuous_comp, continuous_sum_dom, ContinuousAdd.continuous_add, LinearIsometryEquiv.continuous, AddUnits.continuous_embedProduct, continuous_sub_left, ContinuousMap.range_toUniformOnFunIsCompact, continuous_iff_image_closure_subset_closure_image, Metric.Snowflaking.continuous_ofSnowflaking, WithTop.continuous_untop, MulOpposite.continuous_op, intervalIntegral.continuous_primitive, Polynomial.continuous_eval₂, continuous_toMul, Ultrafilter.continuous_mul_left, AddGroupTopology.continuous_neg', ContinuousLinearMapWOT.continuous_dual_apply, continuous_def, continuous_stereoInvFun, continuous_mul_left, IsometryEquiv.continuous, Quaternion.continuous_imK, continuous_empty_function, MeasureTheory.continuous_setIntegral, continuous_of_continuousAt_one₂, continuous_sub_right, Real.continuous_arccos, ContinuousAlternatingMap.coe_continuous, Seminorm.continuous_of_forall, ContinuousAffineMap.continuous, Uniform.continuous_iff'_left, ContinuousMapZero.continuous_toNNReal, continuous_norm, VectorPrebundle.continuous_totalSpaceMk, MeasureTheory.Measure.innerRegularWRT_preimage_one_hasCompactSupport_measure_ne_top_of_addGroup, ContinuousMap.continuous_curry, List.continuous_prod, continuous_skewAdjointPart, continuous_generateFrom_iff, continuous_nndist, EMetric.continuous_infEdist, IsUnit.continuous_const_smul_iff, ENNReal.continuous_div_const, continuous_pow, Real.continuous_arsinh, ContinuousAlternatingMap.continuous_toContinuousMultilinearMap, continuous_max, LocallyFinite.continuous, ContinuousLinearMapWOT.ContinuousLinearMap.continuous_toWOT, Seminorm.continuous_iff_continuous_comp, OnePoint.continuous_iff_from_nat, PartitionOfUnity.continuous_smul, Real.Angle.continuous_cos, Topology.IsScott.scottContinuousOn_iff_continuous, Differentiable.continuous, UpperHalfPlane.continuous_coe, ContMDiff.continuous_tangentMap, ContinuousAddMonoidHom.continuous_toFun, GroupTopology.coinduced_continuous, Preorder.continuous_restrictLe, MvPowerSeries.WithPiTopology.continuous_constantCoeff, RestrictedProduct.continuous_rng_of_bot, ContDiff.continuous_iteratedDeriv', IsCoveringMap.eq_liftPath_iff, MeasureTheory.continuous_L1_toL1, RestrictedProduct.continuous_dom_prod_left, Quaternion.continuous_normSq, Preorder.continuous_frestrictLe₂, Quaternion.continuous_re, LinearMap.mem_span_iff_continuous, GenLoop.continuous_toLoop, NormedAddGroupHom.continuous, continuous_circleMap_inv, Real.fromBinary_continuous, List.continuous_eraseIdx, DiscreteQuotient.map_continuous, continuous_preStoneCechUnit, CantorScheme.VanishingDiam.map_continuous, AbstractCompletion.continuous_extend, continuous_coinduced_rng, MeasureTheory.FiniteMeasure.continuous_lintegral_boundedContinuousFunction, cyclotomicCharacter.continuous, LinearMap.continuous_uncurry_of_isContPerfPair, BarrelledSpace.continuous_of_lowerSemicontinuous, ContinuousOn.comp_fract'', continuous_iff_seqContinuous, unitInterval.continuous_symm, IsTopologicalGroup.continuous_conj, WeakDual.eval_continuous, continuous_iff_continuousAt, continuous_of_mulTSupport, Preorder.continuous_restrictLe₂, Real.continuous_mulExpNegMulSq, continuous_sInf_rng, Topology.IsGeneratedBy.continuous_iff, Complex.continuous_im, IsSpectralMap.toContinuous, ContinuousMultilinearMap.continuous_restrictScalars, Ordinal.isNormal_iff_strictMono_and_continuous, EMetric.continuous_infEdist_hausdorffEdist, Complex.UnitDisc.continuous_pow, MvPolynomial.continuous_eval, ContinuousAffineMap.cont, WithLp.prod_continuous_ofLp, continuous_sigmoid, AffineIsometry.continuous, BoundedContinuousFunction.continuous_comp, Unitization.continuous_inr, Seminorm.cont_normedSpace_to_withSeminorms, FiberBundleCore.continuous_totalSpaceMk, CategoryTheory.PreGaloisCategory.toAut_continuous, contDiff_one_iff_deriv, IsBoundedLinearMap.isLinearMap_and_continuous_iff_isBoundedLinearMap, CompletelyRegularSpace.completely_regular_isOpen, IsTopologicalAddGroup.continuous_addConj_prod, DeltaGeneratedSpace.continuous_iff, ContDiffBump.continuous_normed, IsModuleTopology.continuous_mul_of_finite, MeasureTheory.Lp.continuous_posPart, ApproximatesLinearOn.continuous, AddUnits.continuous_iff, UniformFun.isClosed_setOf_continuous, Topology.IsLowerSet.monotone_to_lowerTopology_continuous, AddUnits.continuous_coe_neg, ContinuousAddMonoidHom.continuous_comp_left, continuous_discrete_rng, IsometryEquiv.comp_continuous_iff, MvPowerSeries.WithPiTopology.continuous_C, AffineMap.continuous_of_finiteDimensional, continuous_iff_continuous_dist, continuous_iff_ultrafilter, WithLp.continuous_snd, Topology.IsEmbedding.continuous, continuous_induced_rng, ContinuousMap.instCanLiftForallCoeContinuous, continuous_ofAdd, MeasureTheory.exists_continuous_eLpNorm_sub_le_of_closed, continuous_nhdsAdjoint_dom, UpperHalfPlane.continuous_re

Theorems

NameKindAssumesProvesValidatesDepends On
closure_image_closure 📖mathematicalContinuousclosure
Set.image
Set.Subset.antisymm
closure_minimal
image_closure_subset_closure_image
isClosed_closure
closure_mono
Set.image_mono
subset_closure
closure_subset_preimage_closure_image 📖mathematicalContinuousSet
Set.instHasSubset
closure
Set.preimage
Set.image
Set.MapsTo.closure
Set.mapsTo_image
continuousAt_congr 📖mathematicalFilter.EventuallyEq
nhds
ContinuousAtFilter.EventuallyEq.eq_of_nhds
Filter.tendsto_congr'
continuousAt_const 📖mathematicalContinuousAttendsto_const_nhds
continuousAt_def 📖mathematicalContinuousAt
Set
Filter
Filter.instMembership
nhds
Set.preimage
continuousAt_id 📖mathematicalContinuousAtContinuous.continuousAt
continuous_id
continuousAt_id' 📖mathematicalContinuousAtcontinuousAt_id
continuous_congr 📖mathematicalContinuous
continuous_const 📖mathematicalContinuouscontinuous_iff_continuousAt
continuousAt_const
continuous_def 📖mathematicalContinuous
IsOpen
Set.preimage
Continuous.isOpen_preimage
continuous_id 📖mathematicalContinuouscontinuous_def
continuous_id' 📖mathematicalContinuouscontinuous_id
continuous_iff_continuousAt 📖mathematicalContinuous
ContinuousAt
Continuous.tendsto
continuous_def
isOpen_iff_mem_nhds
IsOpen.mem_nhds
continuous_iff_image_closure_subset_closure_image 📖mathematicalContinuous
Set
Set.instHasSubset
Set.image
closure
image_closure_subset_closure_image
continuous_iff_isClosed
isClosed_of_closure_subset
Mathlib.Tactic.GCongr.rel_imp_rel
Set.instIsTransSubset
Set.image_subset_iff
subset_refl
Set.instReflSubset
Set.preimage_mono
closure_mono
Set.image_preimage_subset
IsClosed.closure_subset
continuous_iff_isClosed 📖mathematicalContinuous
IsClosed
Set.preimage
continuous_def
Function.Surjective.forall
compl_surjective
continuous_iff_preimage_interior_subset_interior_preimage 📖mathematicalContinuous
Set
Set.instHasSubset
Set.preimage
interior
preimage_interior_subset_interior_preimage
subset_interior_iff_isOpen
Mathlib.Tactic.GCongr.rel_imp_rel
Set.instIsTransSubset
subset_refl
Set.instReflSubset
IsOpen.interior_eq
continuous_of_const 📖mathematicalContinuouscontinuous_iff_continuousAt
Filter.EventuallyEq.continuousAt
Filter.Eventually.of_forall
denseRange_id 📖mathematicalDenseRangeFunction.Surjective.denseRange
denseRange_iff_closure_range 📖mathematicalDenseRange
closure
Set.range
Set.univ
dense_iff_closure_eq
denseRange_subtype_val 📖mathematicalDenseRange
Dense
setOf
Subtype.range_coe_subtype
image_closure_subset_closure_image 📖mathematicalContinuousSet
Set.instHasSubset
Set.image
closure
Set.MapsTo.image_subset
Set.MapsTo.closure
Set.mapsTo_image
map_mem_closure 📖Continuous
Set
Set.instMembership
closure
Set.MapsTo
Set.MapsTo.closure
mem_closure_image 📖mathematicalContinuousAt
Set
Set.instMembership
closure
Set.imagemem_closure_of_frequently_of_tendsto
Filter.Frequently.mono
mem_closure_iff_frequently
Set.mem_image_of_mem
not_continuousAt_of_tendsto 📖mathematicalFilter.Tendsto
Filter
Preorder.toLE
PartialOrder.toPreorder
Filter.instPartialOrder
nhds
Disjoint
BoundedOrder.toOrderBot
CompleteLattice.toBoundedOrder
Filter.instCompleteLatticeFilter
ContinuousAtFilter.Tendsto.not_tendsto
Filter.Tendsto.mono_left
preimage_interior_subset_interior_preimage 📖mathematicalContinuousSet
Set.instHasSubset
Set.preimage
interior
interior_maximal
Set.preimage_mono
interior_subset
IsOpen.preimage
isOpen_interior
tendsto_lift'_closure_nhds 📖mathematicalContinuousFilter.Tendsto
Filter.lift'
nhds
closure
Filter.Tendsto.lift'_closure
Continuous.tendsto

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