Documentation Verification Report

OrderClosed

πŸ“ Source: Mathlib/Topology/Order/OrderClosed.lean

Statistics

MetricCount
DefinitionsClosedIciTopology, ClosedIicTopology, OrderClosedTopology
3
Theoremsclosure, of_closure, closure, of_closure, isClosed_Ici, isClosed_Iic, if_le, max, min, eventually_lt, closure_le, nhdsGE, nhdsGT, nhdsLE, nhdsLT, Iio_eq_biUnion, Ioi_eq_biUnion, exists_between, exists_ge, exists_ge', exists_gt, exists_le, exists_le', exists_lt, orderDual, of_predOrder_succOrder, eventually_const_le, eventually_const_lt, eventually_le_const, eventually_lt, eventually_lt_const, max, max_left, max_right, min, min_left, min_right, tendsto_nhds_max_left, tendsto_nhds_max_right, tendsto_nhds_min_left, tendsto_nhds_min_right, Icc_mem_nhds, Icc_mem_nhdsGE, Icc_mem_nhdsGE_of_mem, Icc_mem_nhdsGT, Icc_mem_nhdsGT_of_mem, Icc_mem_nhdsLE, Icc_mem_nhdsLE_of_mem, Icc_mem_nhdsLT, Icc_mem_nhdsLT_of_mem, Ici_mem_nhds, Ico_mem_nhds, Ico_mem_nhdsGE, Ico_mem_nhdsGE_of_mem, Ico_mem_nhdsGT, Ico_mem_nhdsGT_of_mem, Ico_mem_nhdsLE_of_mem, Ico_mem_nhdsLT, Ico_mem_nhdsLT_of_mem, Iic_mem_nhds, Iio_mem_nhds, Ioc_mem_nhds, Ioc_mem_nhdsGE_of_mem, Ioc_mem_nhdsGT, Ioc_mem_nhdsGT_of_mem, Ioc_mem_nhdsLE, Ioc_mem_nhdsLE_of_mem, Ioc_mem_nhdsLT, Ioc_mem_nhdsLT_of_mem, Ioi_mem_nhds, Ioo_mem_nhds, Ioo_mem_nhdsGE_of_mem, Ioo_mem_nhdsGT, Ioo_mem_nhdsGT_of_mem, Ioo_mem_nhdsLE_of_mem, Ioo_mem_nhdsLT, Ioo_mem_nhdsLT_of_mem, Ioo_subset_closure_interior, epigraph, hypograph, isClosed_le, range_of_tendsto, range_of_tendsto, isClosed_le', to_t2Space, orderClosedTopology', nhdsGE_eq_nhds, nhdsGT_eq_nhdsNE, nhdsLE, nhdsLT, instOrderClosedTopology, nhdsGE, nhdsGT, nhdsLE_eq_nhds, nhdsLT_eq_nhdsNE, bddAbove_closure, bddBelow_closure, closure_Icc, closure_Ici, closure_Iic, closure_le_eq, closure_lt_subset_le, continuousWithinAt_Icc_iff_Ici, continuousWithinAt_Icc_iff_Iic, continuousWithinAt_Ico_iff_Ici, continuousWithinAt_Ico_iff_Iio, continuousWithinAt_Ioc_iff_Iic, continuousWithinAt_Ioc_iff_Ioi, continuousWithinAt_Ioo_iff_Iio, continuousWithinAt_Ioo_iff_Ioi, continuous_if_le, continuous_max, continuous_min, disjoint_nhds_atBot, disjoint_nhds_atBot_iff, disjoint_nhds_atTop, disjoint_nhds_atTop_iff, eventually_ge_nhds, eventually_gt_nhds, eventually_le_nhds, eventually_lt_nhds, frontier_Ici_subset, frontier_Iic_subset, frontier_le_subset_eq, frontier_lt_subset_eq, ge_of_tendsto, ge_of_tendsto', ge_of_tendsto_of_frequently, iInf_eq_of_forall_le_of_tendsto, iSup_eq_of_forall_le_of_tendsto, iUnion_Ici_eq_Ioi_of_lt_of_tendsto, iUnion_Iic_eq_Iio_of_lt_of_tendsto, inf_nhds_atBot, inf_nhds_atTop, instClosedIciTopology, instClosedIciTopologyOrderDual, instClosedIicTopology, instClosedIicTopologyOrderDual, instFirstCountableTopologyOrderDual, instOrderClosedTopologyForall, instOrderClosedTopologyOrderDual, instOrderClosedTopologyProd, instSecondCountableTopologyOrderDual, instSeparableSpaceOrderDual, interior_Iio, interior_Ioi, interior_Ioo, isClosed_Icc, isClosed_Ici, isClosed_Iic, isClosed_antitone, isClosed_antitoneOn, isClosed_le, isClosed_le_prod, isClosed_monotone, isClosed_monotoneOn, isOpen_Iio, isOpen_Ioi, isOpen_Ioo, isOpen_lt, isOpen_lt_prod, le_of_tendsto, le_of_tendsto', le_of_tendsto_of_frequently, le_of_tendsto_of_tendsto, le_of_tendsto_of_tendsto', le_of_tendsto_of_tendsto_of_frequently, le_on_closure, lowerBounds_closure, lt_subset_interior_le, nhdsWithin_Icc_eq_nhdsGE, nhdsWithin_Icc_eq_nhdsLE, nhdsWithin_Ico_eq_nhdsGE, nhdsWithin_Ico_eq_nhdsLT, nhdsWithin_Ioc_eq_nhdsGT, nhdsWithin_Ioc_eq_nhdsLE, nhdsWithin_Ioo_eq_nhdsGT, nhdsWithin_Ioo_eq_nhdsLT, not_tendsto_atBot_of_tendsto_nhds, not_tendsto_atTop_of_tendsto_nhds, not_tendsto_nhds_of_tendsto_atBot, not_tendsto_nhds_of_tendsto_atTop, tendsto_le_of_eventuallyLE, upperBounds_closure
184
Total187

BddAbove

Theorems

NameKindAssumesProvesValidatesDepends On
closure πŸ“–mathematicalBddAbove
Preorder.toLE
closureβ€”bddAbove_closure
of_closure πŸ“–β€”BddAbove
Preorder.toLE
closure
β€”β€”mono
subset_closure

BddBelow

Theorems

NameKindAssumesProvesValidatesDepends On
closure πŸ“–mathematicalBddBelow
Preorder.toLE
closureβ€”bddBelow_closure
of_closure πŸ“–β€”BddBelow
Preorder.toLE
closure
β€”β€”mono
subset_closure

ClosedIciTopology

Theorems

NameKindAssumesProvesValidatesDepends On
isClosed_Ici πŸ“–mathematicalβ€”IsClosed
Set.Ici
β€”β€”

ClosedIicTopology

Theorems

NameKindAssumesProvesValidatesDepends On
isClosed_Iic πŸ“–mathematicalβ€”IsClosed
Set.Iic
β€”β€”

Continuous

Theorems

NameKindAssumesProvesValidatesDepends On
if_le πŸ“–mathematicalContinuousPreorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”continuous_if_le
continuousOn
max πŸ“–mathematicalContinuousSemilatticeSup.toMax
Lattice.toSemilatticeSup
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”min
instOrderClosedTopologyOrderDual
min πŸ“–mathematicalContinuousSemilatticeInf.toMin
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”min_def
if_le

ContinuousAt

Theorems

NameKindAssumesProvesValidatesDepends On
eventually_lt πŸ“–mathematicalContinuousAt
Preorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter.Eventually
nhds
β€”Filter.Tendsto.eventually_lt

ContinuousWithinAt

Theorems

NameKindAssumesProvesValidatesDepends On
closure_le πŸ“–β€”Set
Set.instMembership
closure
ContinuousWithinAt
Preorder.toLE
β€”β€”IsClosed.closure_subset
OrderClosedTopology.isClosed_le'
mem_closure
prodMk

CovBy

Theorems

NameKindAssumesProvesValidatesDepends On
nhdsGE πŸ“–mathematicalCovBy
Preorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
nhdsWithin
Set.Ici
Filter
Filter.instPure
β€”nhdsLE
instClosedIicTopologyOrderDual
toDual
nhdsGT πŸ“–mathematicalCovBy
Preorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
nhdsWithin
Set.Ioi
Bot.bot
Filter
Filter.instBot
β€”nhdsLT
instClosedIicTopologyOrderDual
toDual
nhdsLE πŸ“–mathematicalCovBy
Preorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
nhdsWithin
Set.Iic
Filter
Filter.instPure
β€”Set.Iio_insert
nhdsWithin_insert
nhdsLT
sup_bot_eq
nhdsLT πŸ“–mathematicalCovBy
Preorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
nhdsWithin
Set.Iio
Bot.bot
Filter
Filter.instBot
β€”Filter.empty_mem_iff_bot
Ioo_mem_nhdsLT
Ioo_eq

Dense

Theorems

NameKindAssumesProvesValidatesDepends On
Iio_eq_biUnion πŸ“–mathematicalDenseSet.Iio
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.iUnion
Set
Set.instMembership
Set.instInter
β€”Ioi_eq_biUnion
instOrderClosedTopologyOrderDual
OrderDual.denselyOrdered
Ioi_eq_biUnion πŸ“–mathematicalDenseSet.Ioi
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.iUnion
Set
Set.instMembership
Set.instInter
β€”Set.Subset.antisymm
exists_between
Set.mem_iUnionβ‚‚
Set.iUnionβ‚‚_subset
Set.Ioi_subset_Ioi
le_of_lt
exists_between πŸ“–mathematicalDense
Preorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Set.instMembership
Set.Ioo
β€”exists_mem_open
isOpen_Ioo
Set.nonempty_Ioo
exists_ge πŸ“–mathematicalDenseSet
Set.instMembership
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”LT.lt.le
exists_gt
exists_ge' πŸ“–mathematicalDense
Set
Set.instMembership
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”le_rfl
exists_mem_open
isOpen_Ioi
LT.lt.le
exists_gt πŸ“–mathematicalDenseSet
Set.instMembership
Preorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”exists_mem_open
isOpen_Ioi
NoMaxOrder.exists_gt
exists_le πŸ“–mathematicalDenseSet
Set.instMembership
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”exists_ge
instClosedIicTopologyOrderDual
OrderDual.noMaxOrder
orderDual
exists_le' πŸ“–mathematicalDense
Set
Set.instMembership
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”exists_ge'
instClosedIicTopologyOrderDual
orderDual
exists_lt πŸ“–mathematicalDenseSet
Set.instMembership
Preorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”exists_gt
instClosedIicTopologyOrderDual
OrderDual.noMaxOrder
orderDual
orderDual πŸ“–mathematicalDenseOrderDual
OrderDual.instTopologicalSpace
Set.preimage
DFunLike.coe
Equiv
EquivLike.toFunLike
Equiv.instEquivLike
OrderDual.ofDual
β€”β€”

DiscreteTopology

Theorems

NameKindAssumesProvesValidatesDepends On
of_predOrder_succOrder πŸ“–mathematicalβ€”DiscreteTopologyβ€”discreteTopology_iff_nhds
nhdsWithin_univ
Set.Iic_union_Ioi
nhdsWithin_union
PredOrder.nhdsLE
instClosedIicTopology
SuccOrder.nhdsGT
instClosedIciTopology
sup_bot_eq

Filter

Theorems

NameKindAssumesProvesValidatesDepends On
tendsto_nhds_max_left πŸ“–mathematicalTendsto
nhdsWithin
Set.Ioi
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
SemilatticeSup.toMax
Lattice.toSemilatticeSup
β€”max_comm
tendsto_nhds_max_right
tendsto_nhds_max_right πŸ“–mathematicalTendsto
nhdsWithin
Set.Ioi
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
SemilatticeSup.toMax
Lattice.toSemilatticeSup
β€”tendsto_nhdsWithin_iff
Tendsto.max_right
Eventually.mono
lt_max_of_lt_right
tendsto_nhds_min_left πŸ“–mathematicalTendsto
nhdsWithin
Set.Iio
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
SemilatticeInf.toMinβ€”tendsto_nhds_max_left
instOrderClosedTopologyOrderDual
tendsto_nhds_min_right πŸ“–mathematicalTendsto
nhdsWithin
Set.Iio
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
SemilatticeInf.toMinβ€”tendsto_nhds_max_right
instOrderClosedTopologyOrderDual

Filter.Tendsto

Theorems

NameKindAssumesProvesValidatesDepends On
eventually_const_le πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter.Tendsto
nhds
Filter.Eventually
Preorder.toLE
β€”eventually
eventually_ge_nhds
eventually_const_lt πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter.Tendsto
nhds
Filter.Eventuallyβ€”eventually
eventually_gt_nhds
eventually_le_const πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter.Tendsto
nhds
Filter.Eventually
Preorder.toLE
β€”eventually
eventually_le_nhds
eventually_lt πŸ“–mathematicalFilter.Tendsto
nhds
Preorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter.Eventuallyβ€”LT.lt.exists_disjoint_Iio_Ioi
Filter.Eventually.mp
eventually
Ioi_mem_nhds
instClosedIicTopology
Filter.Eventually.mono
Iio_mem_nhds
instClosedIciTopology
eventually_lt_const πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter.Tendsto
nhds
Filter.Eventuallyβ€”eventually
eventually_lt_nhds
max πŸ“–mathematicalFilter.Tendsto
nhds
SemilatticeSup.toMax
Lattice.toSemilatticeSup
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”comp
Continuous.tendsto
continuous_max
prodMk_nhds
max_left πŸ“–mathematicalFilter.Tendsto
nhds
SemilatticeSup.toMax
Lattice.toSemilatticeSup
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”max_comm
max_right
max_right πŸ“–mathematicalFilter.Tendsto
nhds
SemilatticeSup.toMax
Lattice.toSemilatticeSup
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”sup_idem
max
tendsto_const_nhds
min πŸ“–mathematicalFilter.Tendsto
nhds
SemilatticeInf.toMin
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”comp
Continuous.tendsto
continuous_min
prodMk_nhds
min_left πŸ“–mathematicalFilter.Tendsto
nhds
SemilatticeInf.toMin
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”max_left
instOrderClosedTopologyOrderDual
min_right πŸ“–mathematicalFilter.Tendsto
nhds
SemilatticeInf.toMin
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”max_right
instOrderClosedTopologyOrderDual

IsClosed

Theorems

NameKindAssumesProvesValidatesDepends On
epigraph πŸ“–mathematicalContinuousOnIsClosed
instTopologicalSpaceProd
setOf
Set
Set.instMembership
Preorder.toLE
β€”isClosed_le
preimage
continuous_fst
ContinuousOn.comp
continuousOn_fst
Set.Subset.rfl
continuousOn_snd
hypograph πŸ“–mathematicalContinuousOnIsClosed
instTopologicalSpaceProd
setOf
Set
Set.instMembership
Preorder.toLE
β€”isClosed_le
preimage
continuous_fst
continuousOn_snd
ContinuousOn.comp
continuousOn_fst
Set.Subset.rfl
isClosed_le πŸ“–mathematicalContinuousOnIsClosed
setOf
Set
Set.instMembership
Preorder.toLE
β€”ContinuousOn.preimage_isClosed_of_isClosed
ContinuousOn.prodMk
OrderClosedTopology.isClosed_le'

IsGLB

Theorems

NameKindAssumesProvesValidatesDepends On
range_of_tendsto πŸ“–mathematicalPreorder.toLE
Filter.Tendsto
nhds
IsGLB
Set.range
β€”IsLUB.range_of_tendsto
instClosedIicTopologyOrderDual

IsLUB

Theorems

NameKindAssumesProvesValidatesDepends On
range_of_tendsto πŸ“–mathematicalPreorder.toLE
Filter.Tendsto
nhds
IsLUB
Set.range
β€”Set.forall_mem_range
le_of_tendsto'
Set.mem_range_self

OrderClosedTopology

Theorems

NameKindAssumesProvesValidatesDepends On
isClosed_le' πŸ“–mathematicalβ€”IsClosed
instTopologicalSpaceProd
setOf
Preorder.toLE
β€”β€”
to_t2Space πŸ“–mathematicalβ€”T2Spaceβ€”t2_iff_isClosed_diagonal
IsClosed.inter
isClosed_le'
isClosed_le
continuous_snd
continuous_fst

Pi

Theorems

NameKindAssumesProvesValidatesDepends On
orderClosedTopology' πŸ“–mathematicalβ€”OrderClosedTopology
topologicalSpace
preorder
β€”instOrderClosedTopologyForall

PredOrder

Theorems

NameKindAssumesProvesValidatesDepends On
nhdsGE_eq_nhds πŸ“–mathematicalβ€”nhdsWithin
Set.Ici
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
nhds
β€”nhdsLT_sup_nhdsGE
nhdsLT
bot_sup_eq
nhdsGT_eq_nhdsNE πŸ“–mathematicalβ€”nhdsWithin
Set.Ioi
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Compl.compl
Set
Set.instCompl
Set.instSingletonSet
β€”nhdsLT_sup_nhdsGT
nhdsLT
bot_sup_eq
nhdsLE πŸ“–mathematicalβ€”nhdsWithin
Set.Iic
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter
Filter.instPure
β€”Set.Iio_insert
nhdsWithin_insert
nhdsLT
sup_bot_eq
nhdsLT πŸ“–mathematicalβ€”nhdsWithin
Set.Iio
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Bot.bot
Filter
Filter.instBot
β€”IsMin.Iio_eq
nhdsWithin_empty
CovBy.nhdsLT
Order.pred_covBy_of_not_isMin

Subtype

Theorems

NameKindAssumesProvesValidatesDepends On
instOrderClosedTopology πŸ“–mathematicalβ€”OrderClosedTopology
instTopologicalSpaceSubtype
preorder
β€”Continuous.prodMap
continuous_subtype_val
IsClosed.preimage
OrderClosedTopology.isClosed_le'

SuccOrder

Theorems

NameKindAssumesProvesValidatesDepends On
nhdsGE πŸ“–mathematicalβ€”nhdsWithin
Set.Ici
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter
Filter.instPure
β€”PredOrder.nhdsLE
instClosedIicTopologyOrderDual
nhdsGT πŸ“–mathematicalβ€”nhdsWithin
Set.Ioi
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Bot.bot
Filter
Filter.instBot
β€”PredOrder.nhdsLT
instClosedIicTopologyOrderDual
nhdsLE_eq_nhds πŸ“–mathematicalβ€”nhdsWithin
Set.Iic
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
nhds
β€”PredOrder.nhdsGE_eq_nhds
instClosedIicTopologyOrderDual
nhdsLT_eq_nhdsNE πŸ“–mathematicalβ€”nhdsWithin
Set.Iio
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Compl.compl
Set
Set.instCompl
Set.instSingletonSet
β€”PredOrder.nhdsGT_eq_nhdsNE
instClosedIicTopologyOrderDual

(root)

Definitions

NameCategoryTheorems
ClosedIciTopology πŸ“–CompData
3 mathmath: Topology.IsLower.instClosedIciTopology, instClosedIciTopology, instClosedIciTopologyOrderDual
ClosedIicTopology πŸ“–CompData
4 mathmath: instClosedIicTopology, Topology.IsScott.instClosedIicTopologyOfUnivSet, Topology.IsUpper.instClosedIicTopology, instClosedIicTopologyOrderDual
OrderClosedTopology πŸ“–CompData
11 mathmath: Pi.orderClosedTopology', CStarAlgebra.instOrderClosedTopology, OrderTopology.to_orderClosedTopology, instOrderClosedTopologyOrderDual, Subtype.instOrderClosedTopology, MeasureTheory.Lp.instOrderClosedTopologySubtypeAEEqFunMemAddSubgroupOfClosedIciTopology, HasSolidNorm.orderClosedTopology, RCLike.instOrderClosedTopology, WithZeroTopology.orderClosedTopology, instOrderClosedTopologyProd, Complex.orderClosedTopology

Theorems

NameKindAssumesProvesValidatesDepends On
Icc_mem_nhds πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Filter
Filter.instMembership
nhds
Set.Icc
β€”Filter.mem_of_superset
Ioo_mem_nhds
Set.Ioo_subset_Icc_self
Icc_mem_nhdsGE πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Filter
Filter.instMembership
nhdsWithin
Set.Ici
Set.Icc
β€”Icc_mem_nhdsGE_of_mem
le_rfl
Icc_mem_nhdsGE_of_mem πŸ“–mathematicalSet
Set.instMembership
Set.Ico
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter
Filter.instMembership
nhdsWithin
Set.Ici
Set.Icc
β€”Filter.mem_of_superset
Ico_mem_nhdsGE_of_mem
Set.Ico_subset_Icc_self
Icc_mem_nhdsGT πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Filter
Filter.instMembership
nhdsWithin
Set.Ioi
Set.Icc
β€”Icc_mem_nhdsGT_of_mem
le_rfl
Icc_mem_nhdsGT_of_mem πŸ“–mathematicalSet
Set.instMembership
Set.Ico
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter
Filter.instMembership
nhdsWithin
Set.Ioi
Set.Icc
β€”Filter.mem_of_superset
Ioo_mem_nhdsGT_of_mem
Set.Ioo_subset_Icc_self
Icc_mem_nhdsLE πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Filter
Filter.instMembership
nhdsWithin
Set.Iic
Set.Icc
β€”Icc_mem_nhdsLE_of_mem
le_rfl
Icc_mem_nhdsLE_of_mem πŸ“–mathematicalSet
Set.instMembership
Set.Ioc
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter
Filter.instMembership
nhdsWithin
Set.Iic
Set.Icc
β€”Filter.mem_of_superset
Ioc_mem_nhdsLE_of_mem
Set.Ioc_subset_Icc_self
Icc_mem_nhdsLT πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Filter
Filter.instMembership
nhdsWithin
Set.Iio
Set.Icc
β€”Icc_mem_nhdsLT_of_mem
le_rfl
Icc_mem_nhdsLT_of_mem πŸ“–mathematicalSet
Set.instMembership
Set.Ioc
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter
Filter.instMembership
nhdsWithin
Set.Iio
Set.Icc
β€”Filter.mem_of_superset
Ioo_mem_nhdsLT_of_mem
Set.Ioo_subset_Icc_self
Ici_mem_nhds πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Filter
Filter.instMembership
nhds
Set.Ici
β€”Filter.mem_of_superset
Ioi_mem_nhds
Set.Ioi_subset_Ici_self
Ico_mem_nhds πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Filter
Filter.instMembership
nhds
Set.Ico
β€”Filter.mem_of_superset
Ioo_mem_nhds
Set.Ioo_subset_Ico_self
Ico_mem_nhdsGE πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Filter
Filter.instMembership
nhdsWithin
Set.Ici
Set.Ico
β€”inter_mem_nhdsWithin
Iio_mem_nhds
Ico_mem_nhdsGE_of_mem πŸ“–mathematicalSet
Set.instMembership
Set.Ico
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter
Filter.instMembership
nhdsWithin
Set.Ici
β€”Filter.mem_of_superset
Ico_mem_nhdsGE
Set.Ico_subset_Ico_left
Ico_mem_nhdsGT πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Filter
Filter.instMembership
nhdsWithin
Set.Ioi
Set.Ico
β€”Ico_mem_nhdsGT_of_mem
le_rfl
Ico_mem_nhdsGT_of_mem πŸ“–mathematicalSet
Set.instMembership
Set.Ico
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter
Filter.instMembership
nhdsWithin
Set.Ioi
β€”Filter.mem_of_superset
Ioo_mem_nhdsGT_of_mem
Set.Ioo_subset_Ico_self
Ico_mem_nhdsLE_of_mem πŸ“–mathematicalSet
Set.instMembership
Set.Ioo
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter
Filter.instMembership
nhdsWithin
Set.Iic
Set.Ico
β€”Filter.mem_of_superset
Ioo_mem_nhdsLE_of_mem
Set.Ioo_subset_Ico_self
Ico_mem_nhdsLT πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Filter
Filter.instMembership
nhdsWithin
Set.Iio
Set.Ico
β€”Ico_mem_nhdsLT_of_mem
le_rfl
Ico_mem_nhdsLT_of_mem πŸ“–mathematicalSet
Set.instMembership
Set.Ioc
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter
Filter.instMembership
nhdsWithin
Set.Iio
Set.Ico
β€”Filter.mem_of_superset
Ioo_mem_nhdsLT_of_mem
Set.Ioo_subset_Ico_self
Iic_mem_nhds πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Filter
Filter.instMembership
nhds
Set.Iic
β€”Filter.mem_of_superset
Iio_mem_nhds
Set.Iio_subset_Iic_self
Iio_mem_nhds πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Filter
Filter.instMembership
nhds
Set.Iio
β€”IsOpen.mem_nhds
isOpen_Iio
Ioc_mem_nhds πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Filter
Filter.instMembership
nhds
Set.Ioc
β€”Filter.mem_of_superset
Ioo_mem_nhds
Set.Ioo_subset_Ioc_self
Ioc_mem_nhdsGE_of_mem πŸ“–mathematicalSet
Set.instMembership
Set.Ioo
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter
Filter.instMembership
nhdsWithin
Set.Ici
Set.Ioc
β€”Filter.mem_of_superset
Ioo_mem_nhdsGE_of_mem
Set.Ioo_subset_Ioc_self
Ioc_mem_nhdsGT πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Filter
Filter.instMembership
nhdsWithin
Set.Ioi
Set.Ioc
β€”Ioc_mem_nhdsGT_of_mem
le_rfl
Ioc_mem_nhdsGT_of_mem πŸ“–mathematicalSet
Set.instMembership
Set.Ico
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter
Filter.instMembership
nhdsWithin
Set.Ioi
Set.Ioc
β€”Filter.mem_of_superset
Ioo_mem_nhdsGT_of_mem
Set.Ioo_subset_Ioc_self
Ioc_mem_nhdsLE πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Filter
Filter.instMembership
nhdsWithin
Set.Iic
Set.Ioc
β€”Filter.inter_mem
nhdsWithin_le_nhds
Ioi_mem_nhds
self_mem_nhdsWithin
Ioc_mem_nhdsLE_of_mem πŸ“–mathematicalSet
Set.instMembership
Set.Ioc
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter
Filter.instMembership
nhdsWithin
Set.Iic
β€”Filter.mem_of_superset
Ioc_mem_nhdsLE
Set.Ioc_subset_Ioc_right
Ioc_mem_nhdsLT πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Filter
Filter.instMembership
nhdsWithin
Set.Iio
Set.Ioc
β€”Ioc_mem_nhdsLT_of_mem
le_rfl
Ioc_mem_nhdsLT_of_mem πŸ“–mathematicalSet
Set.instMembership
Set.Ioc
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter
Filter.instMembership
nhdsWithin
Set.Iio
β€”Filter.mem_of_superset
Ioo_mem_nhdsLT_of_mem
Set.Ioo_subset_Ioc_self
Ioi_mem_nhds πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Filter
Filter.instMembership
nhds
Set.Ioi
β€”IsOpen.mem_nhds
isOpen_Ioi
Ioo_mem_nhds πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Filter
Filter.instMembership
nhds
Set.Ioo
β€”IsOpen.mem_nhds
isOpen_Ioo
Ioo_mem_nhdsGE_of_mem πŸ“–mathematicalSet
Set.instMembership
Set.Ioo
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter
Filter.instMembership
nhdsWithin
Set.Ici
β€”Filter.mem_of_superset
Ico_mem_nhdsGE
Set.Ico_subset_Ioo_left
Ioo_mem_nhdsGT πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Filter
Filter.instMembership
nhdsWithin
Set.Ioi
Set.Ioo
β€”Ioo_mem_nhdsGT_of_mem
le_rfl
Ioo_mem_nhdsGT_of_mem πŸ“–mathematicalSet
Set.instMembership
Set.Ico
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter
Filter.instMembership
nhdsWithin
Set.Ioi
Set.Ioo
β€”mem_nhdsWithin
isOpen_Iio
Set.inter_comm
Set.Ioi_inter_Iio
Set.Ioo_subset_Ioo_left
Ioo_mem_nhdsLE_of_mem πŸ“–mathematicalSet
Set.instMembership
Set.Ioo
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter
Filter.instMembership
nhdsWithin
Set.Iic
β€”Filter.mem_of_superset
Ioc_mem_nhdsLE
Set.Ioc_subset_Ioo_right
Ioo_mem_nhdsLT πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set
Filter
Filter.instMembership
nhdsWithin
Set.Iio
Set.Ioo
β€”inter_mem_nhdsWithin
Ioi_mem_nhds
Ioo_mem_nhdsLT_of_mem πŸ“–mathematicalSet
Set.instMembership
Set.Ioc
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter
Filter.instMembership
nhdsWithin
Set.Iio
Set.Ioo
β€”Filter.mem_of_superset
Ioo_mem_nhdsLT
Set.Ioo_subset_Ioo_right
Ioo_subset_closure_interior πŸ“–mathematicalβ€”Set
Set.instHasSubset
Set.Ioo
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
closure
interior
β€”interior_Ioo
bddAbove_closure πŸ“–mathematicalβ€”BddAbove
Preorder.toLE
closure
β€”upperBounds_closure
bddBelow_closure πŸ“–mathematicalβ€”BddBelow
Preorder.toLE
closure
β€”lowerBounds_closure
closure_Icc πŸ“–mathematicalβ€”closure
Set.Icc
β€”IsClosed.closure_eq
isClosed_Icc
closure_Ici πŸ“–mathematicalβ€”closure
Set.Ici
β€”IsClosed.closure_eq
isClosed_Ici
closure_Iic πŸ“–mathematicalβ€”closure
Set.Iic
β€”IsClosed.closure_eq
isClosed_Iic
closure_le_eq πŸ“–mathematicalContinuousclosure
setOf
Preorder.toLE
β€”IsClosed.closure_eq
isClosed_le
closure_lt_subset_le πŸ“–mathematicalContinuousSet
Set.instHasSubset
closure
setOf
Preorder.toLT
Preorder.toLE
β€”closure_minimal
le_of_lt
isClosed_le
continuousWithinAt_Icc_iff_Ici πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousWithinAt
Set.Icc
Set.Ici
β€”nhdsWithin_Icc_eq_nhdsGE
continuousWithinAt_Icc_iff_Iic πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousWithinAt
Set.Icc
Set.Iic
β€”nhdsWithin_Icc_eq_nhdsLE
continuousWithinAt_Ico_iff_Ici πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousWithinAt
Set.Ico
Set.Ici
β€”nhdsWithin_Ico_eq_nhdsGE
continuousWithinAt_Ico_iff_Iio πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousWithinAt
Set.Ico
Set.Iio
β€”nhdsWithin_Ico_eq_nhdsLT
continuousWithinAt_Ioc_iff_Iic πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousWithinAt
Set.Ioc
Set.Iic
β€”nhdsWithin_Ioc_eq_nhdsLE
continuousWithinAt_Ioc_iff_Ioi πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousWithinAt
Set.Ioc
Set.Ioi
β€”nhdsWithin_Ioc_eq_nhdsGT
continuousWithinAt_Ioo_iff_Iio πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousWithinAt
Set.Ioo
Set.Iio
β€”nhdsWithin_Ioo_eq_nhdsLT
continuousWithinAt_Ioo_iff_Ioi πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
ContinuousWithinAt
Set.Ioo
Set.Ioi
β€”nhdsWithin_Ioo_eq_nhdsGT
continuous_if_le πŸ“–β€”Continuous
ContinuousOn
setOf
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”β€”continuous_if
frontier_le_subset_eq
IsClosed.closure_eq
isClosed_le
ContinuousOn.mono
closure_lt_subset_le
continuous_max πŸ“–mathematicalβ€”Continuous
instTopologicalSpaceProd
SemilatticeSup.toMax
Lattice.toSemilatticeSup
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”Continuous.max
continuous_fst
continuous_snd
continuous_min πŸ“–mathematicalβ€”Continuous
instTopologicalSpaceProd
SemilatticeInf.toMin
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”Continuous.min
continuous_fst
continuous_snd
disjoint_nhds_atBot πŸ“–mathematicalβ€”Disjoint
Filter
Filter.instPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
CompleteLattice.toBoundedOrder
Filter.instCompleteLatticeFilter
nhds
Filter.atBot
β€”β€”
disjoint_nhds_atBot_iff πŸ“–mathematicalβ€”Disjoint
Filter
Filter.instPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
CompleteLattice.toBoundedOrder
Filter.instCompleteLatticeFilter
nhds
Filter.atBot
IsBot
β€”mem_of_mem_nhds
Filter.disjoint_principal_right
IsBot.atBot_eq
le_rfl
Filter.disjoint_of_disjoint_of_mem
disjoint_compl_left
IsOpen.mem_nhds
IsClosed.isOpen_compl
isClosed_Iic
Filter.Iic_mem_atBot
disjoint_nhds_atTop πŸ“–mathematicalβ€”Disjoint
Filter
Filter.instPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
CompleteLattice.toBoundedOrder
Filter.instCompleteLatticeFilter
nhds
Filter.atTop
β€”disjoint_nhds_atBot
OrderDual.noBotOrder
instClosedIicTopologyOrderDual
disjoint_nhds_atTop_iff πŸ“–mathematicalβ€”Disjoint
Filter
Filter.instPartialOrder
BoundedOrder.toOrderBot
Preorder.toLE
PartialOrder.toPreorder
CompleteLattice.toBoundedOrder
Filter.instCompleteLatticeFilter
nhds
Filter.atTop
IsTop
β€”disjoint_nhds_atBot_iff
instClosedIicTopologyOrderDual
eventually_ge_nhds πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter.Eventually
Preorder.toLE
nhds
β€”Ici_mem_nhds
eventually_gt_nhds πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter.Eventually
nhds
β€”Ioi_mem_nhds
eventually_le_nhds πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter.Eventually
Preorder.toLE
nhds
β€”Iic_mem_nhds
eventually_lt_nhds πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Filter.Eventually
nhds
β€”Iio_mem_nhds
frontier_Ici_subset πŸ“–mathematicalβ€”Set
Set.instHasSubset
frontier
Set.Ici
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.instSingletonSet
β€”frontier_Iic_subset
instOrderClosedTopologyOrderDual
frontier_Iic_subset πŸ“–mathematicalβ€”Set
Set.instHasSubset
frontier
Set.Iic
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
Set.instSingletonSet
β€”frontier_le_subset_eq
continuous_id
continuous_const
frontier_le_subset_eq πŸ“–mathematicalContinuousSet
Set.instHasSubset
frontier
setOf
Preorder.toLE
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”frontier_eq_closure_inter_closure
closure_le_eq
le_antisymm
closure_lt_subset_le
not_le
frontier_lt_subset_eq πŸ“–mathematicalContinuousSet
Set.instHasSubset
frontier
setOf
Preorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”frontier_compl
frontier_le_subset_eq
ge_of_tendsto πŸ“–β€”Filter.Tendsto
nhds
Filter.Eventually
Preorder.toLE
β€”β€”IsClosed.mem_of_tendsto
isClosed_Ici
ge_of_tendsto' πŸ“–β€”Filter.Tendsto
nhds
Preorder.toLE
β€”β€”ge_of_tendsto
Filter.Eventually.of_forall
ge_of_tendsto_of_frequently πŸ“–β€”Filter.Tendsto
nhds
Filter.Frequently
Preorder.toLE
β€”β€”IsClosed.mem_of_frequently_of_tendsto
isClosed_Ici
iInf_eq_of_forall_le_of_tendsto πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
Filter.Tendsto
nhds
iInf
ConditionallyCompletePartialOrderInf.toInfSet
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderInf
β€”iSup_eq_of_forall_le_of_tendsto
instClosedIicTopologyOrderDual
iSup_eq_of_forall_le_of_tendsto πŸ“–mathematicalPreorder.toLE
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
Filter.Tendsto
nhds
iSup
ConditionallyCompletePartialOrderSup.toSupSet
β€”Filter.nonempty_of_neBot
IsLUB.ciSup_eq
IsLUB.range_of_tendsto
iUnion_Ici_eq_Ioi_of_lt_of_tendsto πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Filter.Tendsto
nhds
Set.iUnion
Set.Ici
Set.Ioi
β€”iUnion_Iic_eq_Iio_of_lt_of_tendsto
instClosedIicTopologyOrderDual
iUnion_Iic_eq_Iio_of_lt_of_tendsto πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
ConditionallyCompletePartialOrderSup.toPartialOrder
ConditionallyCompletePartialOrder.toConditionallyCompletePartialOrderSup
ConditionallyCompleteLattice.toConditionallyCompletePartialOrder
ConditionallyCompleteLinearOrder.toConditionallyCompleteLattice
Filter.Tendsto
nhds
Set.iUnion
Set.Iic
Set.Iio
β€”Set.mem_range
LT.lt.false
Set.biUnion_range
IsLUB.biUnion_Iic_eq_Iio
IsLUB.range_of_tendsto
le_of_lt
inf_nhds_atBot πŸ“–mathematicalβ€”Filter
Filter.instInf
nhds
Filter.atBot
Bot.bot
Filter.instBot
β€”Disjoint.eq_bot
disjoint_nhds_atBot
inf_nhds_atTop πŸ“–mathematicalβ€”Filter
Filter.instInf
nhds
Filter.atTop
Bot.bot
Filter.instBot
β€”Disjoint.eq_bot
disjoint_nhds_atTop
instClosedIciTopology πŸ“–mathematicalβ€”ClosedIciTopologyβ€”isClosed_le
continuous_const
continuous_id
instClosedIciTopologyOrderDual πŸ“–mathematicalβ€”ClosedIciTopology
OrderDual
OrderDual.instTopologicalSpace
OrderDual.instPreorder
β€”isClosed_Iic
instClosedIicTopology πŸ“–mathematicalβ€”ClosedIicTopologyβ€”isClosed_le
continuous_id
continuous_const
instClosedIicTopologyOrderDual πŸ“–mathematicalβ€”ClosedIicTopology
OrderDual
OrderDual.instTopologicalSpace
OrderDual.instPreorder
β€”isClosed_Ici
instFirstCountableTopologyOrderDual πŸ“–mathematicalβ€”FirstCountableTopology
OrderDual
OrderDual.instTopologicalSpace
β€”β€”
instOrderClosedTopologyForall πŸ“–mathematicalOrderClosedTopologyPi.topologicalSpace
Pi.preorder
β€”Set.setOf_forall
isClosed_iInter
isClosed_le
Continuous.fst'
continuous_apply
Continuous.snd'
instOrderClosedTopologyOrderDual πŸ“–mathematicalβ€”OrderClosedTopology
OrderDual
OrderDual.instTopologicalSpace
OrderDual.instPreorder
β€”IsClosed.preimage
continuous_swap
OrderClosedTopology.isClosed_le'
instOrderClosedTopologyProd πŸ“–mathematicalβ€”OrderClosedTopology
instTopologicalSpaceProd
Prod.instPreorder
β€”IsClosed.inter
isClosed_le
Continuous.fst
continuous_fst
continuous_snd
Continuous.snd
instSecondCountableTopologyOrderDual πŸ“–mathematicalβ€”SecondCountableTopology
OrderDual
OrderDual.instTopologicalSpace
β€”β€”
instSeparableSpaceOrderDual πŸ“–mathematicalβ€”TopologicalSpace.SeparableSpace
OrderDual
OrderDual.instTopologicalSpace
β€”β€”
interior_Iio πŸ“–mathematicalβ€”interior
Set.Iio
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”IsOpen.interior_eq
isOpen_Iio
interior_Ioi πŸ“–mathematicalβ€”interior
Set.Ioi
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”IsOpen.interior_eq
isOpen_Ioi
interior_Ioo πŸ“–mathematicalβ€”interior
Set.Ioo
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”IsOpen.interior_eq
isOpen_Ioo
isClosed_Icc πŸ“–mathematicalβ€”IsClosed
Set.Icc
β€”IsClosed.inter
isClosed_Ici
instClosedIciTopology
isClosed_Iic
instClosedIicTopology
isClosed_Ici πŸ“–mathematicalβ€”IsClosed
Set.Ici
β€”ClosedIciTopology.isClosed_Ici
isClosed_Iic πŸ“–mathematicalβ€”IsClosed
Set.Iic
β€”ClosedIicTopology.isClosed_Iic
isClosed_antitone πŸ“–mathematicalβ€”IsClosed
Pi.topologicalSpace
setOf
Antitone
β€”isClosed_monotone
instOrderClosedTopologyOrderDual
isClosed_antitoneOn πŸ“–mathematicalβ€”IsClosed
Pi.topologicalSpace
setOf
AntitoneOn
β€”isClosed_monotoneOn
instOrderClosedTopologyOrderDual
isClosed_le πŸ“–mathematicalContinuousIsClosed
setOf
Preorder.toLE
β€”continuous_iff_isClosed
Continuous.prodMk
isClosed_le_prod
isClosed_le_prod πŸ“–mathematicalβ€”IsClosed
instTopologicalSpaceProd
setOf
Preorder.toLE
β€”OrderClosedTopology.isClosed_le'
isClosed_monotone πŸ“–mathematicalβ€”IsClosed
Pi.topologicalSpace
setOf
Monotone
β€”isClosed_monotoneOn
isClosed_monotoneOn πŸ“–mathematicalβ€”IsClosed
Pi.topologicalSpace
setOf
MonotoneOn
β€”continuousAt_apply
le_of_tendsto_of_tendsto_of_frequently
Filter.Frequently.mono
isOpen_Iio πŸ“–mathematicalβ€”IsOpen
Set.Iio
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”isOpen_Ioi
instClosedIicTopologyOrderDual
isOpen_Ioi πŸ“–mathematicalβ€”IsOpen
Set.Ioi
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”Set.compl_Iic
IsClosed.isOpen_compl
isClosed_Iic
isOpen_Ioo πŸ“–mathematicalβ€”IsOpen
Set.Ioo
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”IsOpen.inter
isOpen_Ioi
instClosedIicTopology
isOpen_Iio
instClosedIciTopology
isOpen_lt πŸ“–mathematicalContinuousIsOpen
setOf
Preorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”IsClosed.isOpen_compl
isClosed_le
isOpen_lt_prod πŸ“–mathematicalβ€”IsOpen
instTopologicalSpaceProd
setOf
Preorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
β€”isOpen_lt
continuous_fst
continuous_snd
le_of_tendsto πŸ“–β€”Filter.Tendsto
nhds
Filter.Eventually
Preorder.toLE
β€”β€”IsClosed.mem_of_tendsto
isClosed_Iic
le_of_tendsto' πŸ“–β€”Filter.Tendsto
nhds
Preorder.toLE
β€”β€”le_of_tendsto
Filter.Eventually.of_forall
le_of_tendsto_of_frequently πŸ“–β€”Filter.Tendsto
nhds
Filter.Frequently
Preorder.toLE
β€”β€”IsClosed.mem_of_frequently_of_tendsto
isClosed_Iic
le_of_tendsto_of_tendsto πŸ“–β€”Filter.Tendsto
nhds
Filter.EventuallyLE
Preorder.toLE
β€”β€”le_of_tendsto_of_tendsto_of_frequently
Filter.Eventually.frequently
le_of_tendsto_of_tendsto' πŸ“–β€”Filter.Tendsto
nhds
Preorder.toLE
β€”β€”le_of_tendsto_of_tendsto
Filter.Eventually.of_forall
le_of_tendsto_of_tendsto_of_frequently πŸ“–β€”Filter.Tendsto
nhds
Filter.Frequently
Preorder.toLE
β€”β€”IsClosed.mem_of_frequently_of_tendsto
OrderClosedTopology.isClosed_le'
Filter.Tendsto.prodMk_nhds
le_on_closure πŸ“–β€”Preorder.toLE
ContinuousOn
closure
Set
Set.instMembership
β€”β€”subset_closure
closure_minimal
IsClosed.isClosed_le
isClosed_closure
lowerBounds_closure πŸ“–mathematicalβ€”lowerBounds
Preorder.toLE
closure
β€”Set.ext
IsClosed.closure_subset_iff
isClosed_Ici
lt_subset_interior_le πŸ“–mathematicalContinuousSet
Set.instHasSubset
setOf
Preorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
interior
Preorder.toLE
β€”interior_maximal
le_of_lt
isOpen_lt
nhdsWithin_Icc_eq_nhdsGE πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
nhdsWithin
Set.Icc
Set.Ici
β€”nhdsWithin_inter_of_mem'
nhdsWithin_le_nhds
Iic_mem_nhds
nhdsWithin_Icc_eq_nhdsLE πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
nhdsWithin
Set.Icc
Set.Iic
β€”nhdsWithin_inter_of_mem
nhdsWithin_le_nhds
Ici_mem_nhds
nhdsWithin_Ico_eq_nhdsGE πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
nhdsWithin
Set.Ico
Set.Ici
β€”nhdsWithin_inter_of_mem'
nhdsWithin_le_nhds
Iio_mem_nhds
nhdsWithin_Ico_eq_nhdsLT πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
nhdsWithin
Set.Ico
Set.Iio
β€”nhdsWithin_inter_of_mem
nhdsWithin_le_nhds
Ici_mem_nhds
nhdsWithin_Ioc_eq_nhdsGT πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
nhdsWithin
Set.Ioc
Set.Ioi
β€”nhdsWithin_inter_of_mem'
nhdsWithin_le_nhds
Iic_mem_nhds
nhdsWithin_Ioc_eq_nhdsLE πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
nhdsWithin
Set.Ioc
Set.Iic
β€”nhdsWithin_inter_of_mem
nhdsWithin_le_nhds
Ioi_mem_nhds
nhdsWithin_Ioo_eq_nhdsGT πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
nhdsWithin
Set.Ioo
Set.Ioi
β€”nhdsWithin_inter_of_mem'
nhdsWithin_le_nhds
Iio_mem_nhds
nhdsWithin_Ioo_eq_nhdsLT πŸ“–mathematicalPreorder.toLT
PartialOrder.toPreorder
SemilatticeInf.toPartialOrder
Lattice.toSemilatticeInf
DistribLattice.toLattice
instDistribLatticeOfLinearOrder
nhdsWithin
Set.Ioo
Set.Iio
β€”nhdsWithin_inter_of_mem
nhdsWithin_le_nhds
Ioi_mem_nhds
not_tendsto_atBot_of_tendsto_nhds πŸ“–mathematicalFilter.Tendsto
nhds
Filter.atBotβ€”Filter.Tendsto.not_tendsto
disjoint_nhds_atBot
not_tendsto_atTop_of_tendsto_nhds πŸ“–mathematicalFilter.Tendsto
nhds
Filter.atTopβ€”Filter.Tendsto.not_tendsto
disjoint_nhds_atTop
not_tendsto_nhds_of_tendsto_atBot πŸ“–mathematicalFilter.Tendsto
Filter.atBot
nhdsβ€”Filter.Tendsto.not_tendsto
Disjoint.symm
disjoint_nhds_atBot
not_tendsto_nhds_of_tendsto_atTop πŸ“–mathematicalFilter.Tendsto
Filter.atTop
nhdsβ€”Filter.Tendsto.not_tendsto
Disjoint.symm
disjoint_nhds_atTop
tendsto_le_of_eventuallyLE πŸ“–β€”Filter.Tendsto
nhds
Filter.EventuallyLE
Preorder.toLE
β€”β€”le_of_tendsto_of_tendsto
upperBounds_closure πŸ“–mathematicalβ€”upperBounds
Preorder.toLE
closure
β€”Set.ext
IsClosed.closure_subset_iff
isClosed_Iic

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