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Mathlib.Topology.CWComplex.Classical.Finite

Finiteness notions on CW complexes #

In this file we define what it means for a CW complex to be finite dimensional, of finite type or finite. We define constructors with relaxed conditions for CW complexes of finite type and finite CW complexes.

Main definitions #

Main statements #

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class Topology.RelCWComplex.FiniteDimensional {X : Type u} [TopologicalSpace X] (C : Set X) {D : Set X} [RelCWComplex C D] :
Prop

A CW complex is finite dimensional if cell C n is empty for all but finitely many n.

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    class Topology.RelCWComplex.FiniteType {X : Type u} [TopologicalSpace X] (C : Set X) {D : Set X} [RelCWComplex C D] :
    Prop

    A CW complex is of finite type if cell C n is finite for every n.

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      class Topology.RelCWComplex.Finite {X : Type u_1} [TopologicalSpace X] (C : Set X) {D : Set X} [RelCWComplex C D] extends Topology.RelCWComplex.FiniteDimensional C, Topology.RelCWComplex.FiniteType C :
      Prop

      A CW complex is finite if it is finite dimensional and of finite type.

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        theorem Topology.RelCWComplex.finite_of_finiteDimensional_finiteType {X : Type u_1} [TopologicalSpace X] (C : Set X) {D : Set X} [RelCWComplex C D] [FiniteDimensional C] [FiniteType C] :
        Finite C
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        def Topology.RelCWComplex.mkFiniteType {X : Type u} [TopologicalSpace X] (C : Set X) (D : outParam (Set X)) (cell : โ„• โ†’ Type u) (map : (n : โ„•) โ†’ cell n โ†’ PartialEquiv (Fin n โ†’ โ„) X) (finite_cell : โˆ€ (n : โ„•), _root_.Finite (cell n)) (source_eq : โˆ€ (n : โ„•) (i : cell n), (map n i).source = Metric.ball 0 1) (continuousOn : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i)) (Metric.closedBall 0 1)) (continuousOn_symm : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i).symm) (map n i).target) (pairwiseDisjoint' : Set.univ.PairwiseDisjoint fun (ni : (n : โ„•) ร— cell n) => โ†‘(map ni.fst ni.snd) '' Metric.ball 0 1) (disjointBase' : โˆ€ (n : โ„•) (i : cell n), Disjoint (โ†‘(map n i) '' Metric.ball 0 1) D) (mapsTo : โˆ€ (n : โ„•) (i : cell n), Set.MapsTo (โ†‘(map n i)) (Metric.sphere 0 1) (D โˆช โ‹ƒ (m : โ„•), โ‹ƒ (_ : m < n), โ‹ƒ (j : cell m), โ†‘(map m j) '' Metric.closedBall 0 1)) (closed' : โˆ€ A โŠ† C, (โˆ€ (n : โ„•) (j : cell n), IsClosed (A โˆฉ โ†‘(map n j) '' Metric.closedBall 0 1)) โˆง IsClosed (A โˆฉ D) โ†’ IsClosed A) (isClosedBase : IsClosed D) (union' : D โˆช โ‹ƒ (n : โ„•), โ‹ƒ (j : cell n), โ†‘(map n j) '' Metric.closedBall 0 1 = C) :
        RelCWComplex C D

        If we want to construct a relative CW complex of finite type, we can add the condition finite_cell and relax the condition mapsTo.

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            theorem Topology.RelCWComplex.mkFiniteType_map {X : Type u} [TopologicalSpace X] (C : Set X) (D : outParam (Set X)) (cell : โ„• โ†’ Type u) (map : (n : โ„•) โ†’ cell n โ†’ PartialEquiv (Fin n โ†’ โ„) X) (finite_cell : โˆ€ (n : โ„•), _root_.Finite (cell n)) (source_eq : โˆ€ (n : โ„•) (i : cell n), (map n i).source = Metric.ball 0 1) (continuousOn : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i)) (Metric.closedBall 0 1)) (continuousOn_symm : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i).symm) (map n i).target) (pairwiseDisjoint' : Set.univ.PairwiseDisjoint fun (ni : (n : โ„•) ร— cell n) => โ†‘(map ni.fst ni.snd) '' Metric.ball 0 1) (disjointBase' : โˆ€ (n : โ„•) (i : cell n), Disjoint (โ†‘(map n i) '' Metric.ball 0 1) D) (mapsTo : โˆ€ (n : โ„•) (i : cell n), Set.MapsTo (โ†‘(map n i)) (Metric.sphere 0 1) (D โˆช โ‹ƒ (m : โ„•), โ‹ƒ (_ : m < n), โ‹ƒ (j : cell m), โ†‘(map m j) '' Metric.closedBall 0 1)) (closed' : โˆ€ A โŠ† C, (โˆ€ (n : โ„•) (j : cell n), IsClosed (A โˆฉ โ†‘(map n j) '' Metric.closedBall 0 1)) โˆง IsClosed (A โˆฉ D) โ†’ IsClosed A) (isClosedBase : IsClosed D) (union' : D โˆช โ‹ƒ (n : โ„•), โ‹ƒ (j : cell n), โ†‘(map n j) '' Metric.closedBall 0 1 = C) (n : โ„•) (i : cell n) :
            RelCWComplex.map n i = map n i
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            theorem Topology.RelCWComplex.mkFiniteType_cell {X : Type u} [TopologicalSpace X] (C : Set X) (D : outParam (Set X)) (cell : โ„• โ†’ Type u) (map : (n : โ„•) โ†’ cell n โ†’ PartialEquiv (Fin n โ†’ โ„) X) (finite_cell : โˆ€ (n : โ„•), _root_.Finite (cell n)) (source_eq : โˆ€ (n : โ„•) (i : cell n), (map n i).source = Metric.ball 0 1) (continuousOn : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i)) (Metric.closedBall 0 1)) (continuousOn_symm : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i).symm) (map n i).target) (pairwiseDisjoint' : Set.univ.PairwiseDisjoint fun (ni : (n : โ„•) ร— cell n) => โ†‘(map ni.fst ni.snd) '' Metric.ball 0 1) (disjointBase' : โˆ€ (n : โ„•) (i : cell n), Disjoint (โ†‘(map n i) '' Metric.ball 0 1) D) (mapsTo : โˆ€ (n : โ„•) (i : cell n), Set.MapsTo (โ†‘(map n i)) (Metric.sphere 0 1) (D โˆช โ‹ƒ (m : โ„•), โ‹ƒ (_ : m < n), โ‹ƒ (j : cell m), โ†‘(map m j) '' Metric.closedBall 0 1)) (closed' : โˆ€ A โŠ† C, (โˆ€ (n : โ„•) (j : cell n), IsClosed (A โˆฉ โ†‘(map n j) '' Metric.closedBall 0 1)) โˆง IsClosed (A โˆฉ D) โ†’ IsClosed A) (isClosedBase : IsClosed D) (union' : D โˆช โ‹ƒ (n : โ„•), โ‹ƒ (j : cell n), โ†‘(map n j) '' Metric.closedBall 0 1 = C) (n : โ„•) :
            RelCWComplex.cell C n = cell n
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            theorem Topology.RelCWComplex.finiteType_mkFiniteType {X : Type u} [TopologicalSpace X] (C : Set X) (D : outParam (Set X)) (cell : โ„• โ†’ Type u) (map : (n : โ„•) โ†’ cell n โ†’ PartialEquiv (Fin n โ†’ โ„) X) (finite_cell : โˆ€ (n : โ„•), _root_.Finite (cell n)) (source_eq : โˆ€ (n : โ„•) (i : cell n), (map n i).source = Metric.ball 0 1) (continuousOn : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i)) (Metric.closedBall 0 1)) (continuousOn_symm : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i).symm) (map n i).target) (pairwiseDisjoint' : Set.univ.PairwiseDisjoint fun (ni : (n : โ„•) ร— cell n) => โ†‘(map ni.fst ni.snd) '' Metric.ball 0 1) (disjointBase' : โˆ€ (n : โ„•) (i : cell n), Disjoint (โ†‘(map n i) '' Metric.ball 0 1) D) (mapsTo : โˆ€ (n : โ„•) (i : cell n), Set.MapsTo (โ†‘(map n i)) (Metric.sphere 0 1) (D โˆช โ‹ƒ (m : โ„•), โ‹ƒ (_ : m < n), โ‹ƒ (j : cell m), โ†‘(map m j) '' Metric.closedBall 0 1)) (closed' : โˆ€ A โŠ† C, (โˆ€ (n : โ„•) (j : cell n), IsClosed (A โˆฉ โ†‘(map n j) '' Metric.closedBall 0 1)) โˆง IsClosed (A โˆฉ D) โ†’ IsClosed A) (isClosedBase : IsClosed D) (union' : D โˆช โ‹ƒ (n : โ„•), โ‹ƒ (j : cell n), โ†‘(map n j) '' Metric.closedBall 0 1 = C) :
            FiniteType C

            A CW complex that was constructed using RelCWComplex.mkFiniteType is of finite type.

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            def Topology.CWComplex.mkFiniteType {X : Type u} [TopologicalSpace X] (C : Set X) (cell : โ„• โ†’ Type u) (map : (n : โ„•) โ†’ cell n โ†’ PartialEquiv (Fin n โ†’ โ„) X) (finite_cell : โˆ€ (n : โ„•), _root_.Finite (cell n)) (source_eq : โˆ€ (n : โ„•) (i : cell n), (map n i).source = Metric.ball 0 1) (continuousOn : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i)) (Metric.closedBall 0 1)) (continuousOn_symm : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i).symm) (map n i).target) (pairwiseDisjoint' : Set.univ.PairwiseDisjoint fun (ni : (n : โ„•) ร— cell n) => โ†‘(map ni.fst ni.snd) '' Metric.ball 0 1) (mapsTo : โˆ€ (n : โ„•) (i : cell n), Set.MapsTo (โ†‘(map n i)) (Metric.sphere 0 1) (โ‹ƒ (m : โ„•), โ‹ƒ (_ : m < n), โ‹ƒ (j : cell m), โ†‘(map m j) '' Metric.closedBall 0 1)) (closed' : โˆ€ A โŠ† C, (โˆ€ (n : โ„•) (j : cell n), IsClosed (A โˆฉ โ†‘(map n j) '' Metric.closedBall 0 1)) โ†’ IsClosed A) (union' : โ‹ƒ (n : โ„•), โ‹ƒ (j : cell n), โ†‘(map n j) '' Metric.closedBall 0 1 = C) :
            CWComplex C

            If we want to construct a CW complex of finite type, we can add the condition finite_cell and relax the condition mapsTo.

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                theorem Topology.CWComplex.mkFiniteType_cell {X : Type u} [TopologicalSpace X] (C : Set X) (cell : โ„• โ†’ Type u) (map : (n : โ„•) โ†’ cell n โ†’ PartialEquiv (Fin n โ†’ โ„) X) (finite_cell : โˆ€ (n : โ„•), _root_.Finite (cell n)) (source_eq : โˆ€ (n : โ„•) (i : cell n), (map n i).source = Metric.ball 0 1) (continuousOn : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i)) (Metric.closedBall 0 1)) (continuousOn_symm : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i).symm) (map n i).target) (pairwiseDisjoint' : Set.univ.PairwiseDisjoint fun (ni : (n : โ„•) ร— cell n) => โ†‘(map ni.fst ni.snd) '' Metric.ball 0 1) (mapsTo : โˆ€ (n : โ„•) (i : cell n), Set.MapsTo (โ†‘(map n i)) (Metric.sphere 0 1) (โ‹ƒ (m : โ„•), โ‹ƒ (_ : m < n), โ‹ƒ (j : cell m), โ†‘(map m j) '' Metric.closedBall 0 1)) (closed' : โˆ€ A โŠ† C, (โˆ€ (n : โ„•) (j : cell n), IsClosed (A โˆฉ โ†‘(map n j) '' Metric.closedBall 0 1)) โ†’ IsClosed A) (union' : โ‹ƒ (n : โ„•), โ‹ƒ (j : cell n), โ†‘(map n j) '' Metric.closedBall 0 1 = C) (n : โ„•) :
                Topology.CWComplex.cell C n = cell n
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                theorem Topology.CWComplex.mkFiniteType_map {X : Type u} [TopologicalSpace X] (C : Set X) (cell : โ„• โ†’ Type u) (map : (n : โ„•) โ†’ cell n โ†’ PartialEquiv (Fin n โ†’ โ„) X) (finite_cell : โˆ€ (n : โ„•), _root_.Finite (cell n)) (source_eq : โˆ€ (n : โ„•) (i : cell n), (map n i).source = Metric.ball 0 1) (continuousOn : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i)) (Metric.closedBall 0 1)) (continuousOn_symm : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i).symm) (map n i).target) (pairwiseDisjoint' : Set.univ.PairwiseDisjoint fun (ni : (n : โ„•) ร— cell n) => โ†‘(map ni.fst ni.snd) '' Metric.ball 0 1) (mapsTo : โˆ€ (n : โ„•) (i : cell n), Set.MapsTo (โ†‘(map n i)) (Metric.sphere 0 1) (โ‹ƒ (m : โ„•), โ‹ƒ (_ : m < n), โ‹ƒ (j : cell m), โ†‘(map m j) '' Metric.closedBall 0 1)) (closed' : โˆ€ A โŠ† C, (โˆ€ (n : โ„•) (j : cell n), IsClosed (A โˆฉ โ†‘(map n j) '' Metric.closedBall 0 1)) โ†’ IsClosed A) (union' : โ‹ƒ (n : โ„•), โ‹ƒ (j : cell n), โ†‘(map n j) '' Metric.closedBall 0 1 = C) (n : โ„•) (i : cell n) :
                Topology.CWComplex.map n i = map n i
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                theorem Topology.CWComplex.finiteType_mkFiniteType {X : Type u} [TopologicalSpace X] (C : Set X) (cell : โ„• โ†’ Type u) (map : (n : โ„•) โ†’ cell n โ†’ PartialEquiv (Fin n โ†’ โ„) X) (finite_cell : โˆ€ (n : โ„•), _root_.Finite (cell n)) (source_eq : โˆ€ (n : โ„•) (i : cell n), (map n i).source = Metric.ball 0 1) (continuousOn : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i)) (Metric.closedBall 0 1)) (continuousOn_symm : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i).symm) (map n i).target) (pairwiseDisjoint' : Set.univ.PairwiseDisjoint fun (ni : (n : โ„•) ร— cell n) => โ†‘(map ni.fst ni.snd) '' Metric.ball 0 1) (mapsTo : โˆ€ (n : โ„•) (i : cell n), Set.MapsTo (โ†‘(map n i)) (Metric.sphere 0 1) (โ‹ƒ (m : โ„•), โ‹ƒ (_ : m < n), โ‹ƒ (j : cell m), โ†‘(map m j) '' Metric.closedBall 0 1)) (closed' : โˆ€ A โŠ† C, (โˆ€ (n : โ„•) (j : cell n), IsClosed (A โˆฉ โ†‘(map n j) '' Metric.closedBall 0 1)) โ†’ IsClosed A) (union' : โ‹ƒ (n : โ„•), โ‹ƒ (j : cell n), โ†‘(map n j) '' Metric.closedBall 0 1 = C) :
                FiniteType C

                A CW complex that was constructed using CWComplex.mkFiniteType is of finite type.

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                def Topology.RelCWComplex.mkFinite {X : Type u} [TopologicalSpace X] (C : Set X) (D : outParam (Set X)) (cell : โ„• โ†’ Type u) (map : (n : โ„•) โ†’ cell n โ†’ PartialEquiv (Fin n โ†’ โ„) X) (eventually_isEmpty_cell : โˆ€แถ  (n : โ„•) in Filter.atTop, IsEmpty (cell n)) (finite_cell : โˆ€ (n : โ„•), _root_.Finite (cell n)) (source_eq : โˆ€ (n : โ„•) (i : cell n), (map n i).source = Metric.ball 0 1) (continuousOn : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i)) (Metric.closedBall 0 1)) (continuousOn_symm : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i).symm) (map n i).target) (pairwiseDisjoint' : Set.univ.PairwiseDisjoint fun (ni : (n : โ„•) ร— cell n) => โ†‘(map ni.fst ni.snd) '' Metric.ball 0 1) (disjointBase' : โˆ€ (n : โ„•) (i : cell n), Disjoint (โ†‘(map n i) '' Metric.ball 0 1) D) (mapsTo : โˆ€ (n : โ„•) (i : cell n), Set.MapsTo (โ†‘(map n i)) (Metric.sphere 0 1) (D โˆช โ‹ƒ (m : โ„•), โ‹ƒ (_ : m < n), โ‹ƒ (j : cell m), โ†‘(map m j) '' Metric.closedBall 0 1)) (isClosedBase : IsClosed D) (union' : D โˆช โ‹ƒ (n : โ„•), โ‹ƒ (j : cell n), โ†‘(map n j) '' Metric.closedBall 0 1 = C) :
                RelCWComplex C D

                If we want to construct a finite relative CW complex we can add the conditions eventually_isEmpty_cell and finite_cell, relax the condition mapsTo and remove the condition closed'.

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                    theorem Topology.RelCWComplex.mkFinite_map {X : Type u} [TopologicalSpace X] (C : Set X) (D : outParam (Set X)) (cell : โ„• โ†’ Type u) (map : (n : โ„•) โ†’ cell n โ†’ PartialEquiv (Fin n โ†’ โ„) X) (eventually_isEmpty_cell : โˆ€แถ  (n : โ„•) in Filter.atTop, IsEmpty (cell n)) (finite_cell : โˆ€ (n : โ„•), _root_.Finite (cell n)) (source_eq : โˆ€ (n : โ„•) (i : cell n), (map n i).source = Metric.ball 0 1) (continuousOn : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i)) (Metric.closedBall 0 1)) (continuousOn_symm : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i).symm) (map n i).target) (pairwiseDisjoint' : Set.univ.PairwiseDisjoint fun (ni : (n : โ„•) ร— cell n) => โ†‘(map ni.fst ni.snd) '' Metric.ball 0 1) (disjointBase' : โˆ€ (n : โ„•) (i : cell n), Disjoint (โ†‘(map n i) '' Metric.ball 0 1) D) (mapsTo : โˆ€ (n : โ„•) (i : cell n), Set.MapsTo (โ†‘(map n i)) (Metric.sphere 0 1) (D โˆช โ‹ƒ (m : โ„•), โ‹ƒ (_ : m < n), โ‹ƒ (j : cell m), โ†‘(map m j) '' Metric.closedBall 0 1)) (isClosedBase : IsClosed D) (union' : D โˆช โ‹ƒ (n : โ„•), โ‹ƒ (j : cell n), โ†‘(map n j) '' Metric.closedBall 0 1 = C) (n : โ„•) (i : cell n) :
                    RelCWComplex.map n i = map n i
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                    theorem Topology.RelCWComplex.mkFinite_cell {X : Type u} [TopologicalSpace X] (C : Set X) (D : outParam (Set X)) (cell : โ„• โ†’ Type u) (map : (n : โ„•) โ†’ cell n โ†’ PartialEquiv (Fin n โ†’ โ„) X) (eventually_isEmpty_cell : โˆ€แถ  (n : โ„•) in Filter.atTop, IsEmpty (cell n)) (finite_cell : โˆ€ (n : โ„•), _root_.Finite (cell n)) (source_eq : โˆ€ (n : โ„•) (i : cell n), (map n i).source = Metric.ball 0 1) (continuousOn : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i)) (Metric.closedBall 0 1)) (continuousOn_symm : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i).symm) (map n i).target) (pairwiseDisjoint' : Set.univ.PairwiseDisjoint fun (ni : (n : โ„•) ร— cell n) => โ†‘(map ni.fst ni.snd) '' Metric.ball 0 1) (disjointBase' : โˆ€ (n : โ„•) (i : cell n), Disjoint (โ†‘(map n i) '' Metric.ball 0 1) D) (mapsTo : โˆ€ (n : โ„•) (i : cell n), Set.MapsTo (โ†‘(map n i)) (Metric.sphere 0 1) (D โˆช โ‹ƒ (m : โ„•), โ‹ƒ (_ : m < n), โ‹ƒ (j : cell m), โ†‘(map m j) '' Metric.closedBall 0 1)) (isClosedBase : IsClosed D) (union' : D โˆช โ‹ƒ (n : โ„•), โ‹ƒ (j : cell n), โ†‘(map n j) '' Metric.closedBall 0 1 = C) (n : โ„•) :
                    RelCWComplex.cell C n = cell n
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                    ๐Ÿ“ coverage
                    theorem Topology.RelCWComplex.finite_mkFinite {X : Type u} [TopologicalSpace X] (C : Set X) (D : outParam (Set X)) (cell : โ„• โ†’ Type u) (map : (n : โ„•) โ†’ cell n โ†’ PartialEquiv (Fin n โ†’ โ„) X) (eventually_isEmpty_cell : โˆ€แถ  (n : โ„•) in Filter.atTop, IsEmpty (cell n)) (finite_cell : โˆ€ (n : โ„•), _root_.Finite (cell n)) (source_eq : โˆ€ (n : โ„•) (i : cell n), (map n i).source = Metric.ball 0 1) (continuousOn : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i)) (Metric.closedBall 0 1)) (continuousOn_symm : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i).symm) (map n i).target) (pairwiseDisjoint' : Set.univ.PairwiseDisjoint fun (ni : (n : โ„•) ร— cell n) => โ†‘(map ni.fst ni.snd) '' Metric.ball 0 1) (disjointBase' : โˆ€ (n : โ„•) (i : cell n), Disjoint (โ†‘(map n i) '' Metric.ball 0 1) D) (mapsTo : โˆ€ (n : โ„•) (i : cell n), Set.MapsTo (โ†‘(map n i)) (Metric.sphere 0 1) (D โˆช โ‹ƒ (m : โ„•), โ‹ƒ (_ : m < n), โ‹ƒ (j : cell m), โ†‘(map m j) '' Metric.closedBall 0 1)) (isClosedBase : IsClosed D) (union' : D โˆช โ‹ƒ (n : โ„•), โ‹ƒ (j : cell n), โ†‘(map n j) '' Metric.closedBall 0 1 = C) :
                    Finite C

                    A CW complex that was constructed using RelCWComplex.mkFinite is finite.

                    source
                    ๐Ÿ”ธ coverage
                    def Topology.CWComplex.mkFinite {X : Type u} [TopologicalSpace X] (C : Set X) (cell : โ„• โ†’ Type u) (map : (n : โ„•) โ†’ cell n โ†’ PartialEquiv (Fin n โ†’ โ„) X) (eventually_isEmpty_cell : โˆ€แถ  (n : โ„•) in Filter.atTop, IsEmpty (cell n)) (finite_cell : โˆ€ (n : โ„•), _root_.Finite (cell n)) (source_eq : โˆ€ (n : โ„•) (i : cell n), (map n i).source = Metric.ball 0 1) (continuousOn : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i)) (Metric.closedBall 0 1)) (continuousOn_symm : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i).symm) (map n i).target) (pairwiseDisjoint' : Set.univ.PairwiseDisjoint fun (ni : (n : โ„•) ร— cell n) => โ†‘(map ni.fst ni.snd) '' Metric.ball 0 1) (mapsTo_iff_image_subset : โˆ€ (n : โ„•) (i : cell n), Set.MapsTo (โ†‘(map n i)) (Metric.sphere 0 1) (โ‹ƒ (m : โ„•), โ‹ƒ (_ : m < n), โ‹ƒ (j : cell m), โ†‘(map m j) '' Metric.closedBall 0 1)) (union' : โ‹ƒ (n : โ„•), โ‹ƒ (j : cell n), โ†‘(map n j) '' Metric.closedBall 0 1 = C) :
                    CWComplex C

                    If we want to construct a finite CW complex we can add the conditions eventually_isEmpty_cell and finite_cell, relax the condition mapsTo and remove the condition closed'.

                    Equations
                      Instances For
                        source
                        ๐Ÿ“ coverage
                        theorem Topology.CWComplex.mkFinite_cell {X : Type u} [TopologicalSpace X] (C : Set X) (cell : โ„• โ†’ Type u) (map : (n : โ„•) โ†’ cell n โ†’ PartialEquiv (Fin n โ†’ โ„) X) (eventually_isEmpty_cell : โˆ€แถ  (n : โ„•) in Filter.atTop, IsEmpty (cell n)) (finite_cell : โˆ€ (n : โ„•), _root_.Finite (cell n)) (source_eq : โˆ€ (n : โ„•) (i : cell n), (map n i).source = Metric.ball 0 1) (continuousOn : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i)) (Metric.closedBall 0 1)) (continuousOn_symm : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i).symm) (map n i).target) (pairwiseDisjoint' : Set.univ.PairwiseDisjoint fun (ni : (n : โ„•) ร— cell n) => โ†‘(map ni.fst ni.snd) '' Metric.ball 0 1) (mapsTo_iff_image_subset : โˆ€ (n : โ„•) (i : cell n), Set.MapsTo (โ†‘(map n i)) (Metric.sphere 0 1) (โ‹ƒ (m : โ„•), โ‹ƒ (_ : m < n), โ‹ƒ (j : cell m), โ†‘(map m j) '' Metric.closedBall 0 1)) (union' : โ‹ƒ (n : โ„•), โ‹ƒ (j : cell n), โ†‘(map n j) '' Metric.closedBall 0 1 = C) (n : โ„•) :
                        Topology.CWComplex.cell C n = RelCWComplex.cell C n
                        source
                        ๐Ÿ“ coverage
                        theorem Topology.CWComplex.mkFinite_map {X : Type u} [TopologicalSpace X] (C : Set X) (cell : โ„• โ†’ Type u) (map : (n : โ„•) โ†’ cell n โ†’ PartialEquiv (Fin n โ†’ โ„) X) (eventually_isEmpty_cell : โˆ€แถ  (n : โ„•) in Filter.atTop, IsEmpty (cell n)) (finite_cell : โˆ€ (n : โ„•), _root_.Finite (cell n)) (source_eq : โˆ€ (n : โ„•) (i : cell n), (map n i).source = Metric.ball 0 1) (continuousOn : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i)) (Metric.closedBall 0 1)) (continuousOn_symm : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i).symm) (map n i).target) (pairwiseDisjoint' : Set.univ.PairwiseDisjoint fun (ni : (n : โ„•) ร— cell n) => โ†‘(map ni.fst ni.snd) '' Metric.ball 0 1) (mapsTo_iff_image_subset : โˆ€ (n : โ„•) (i : cell n), Set.MapsTo (โ†‘(map n i)) (Metric.sphere 0 1) (โ‹ƒ (m : โ„•), โ‹ƒ (_ : m < n), โ‹ƒ (j : cell m), โ†‘(map m j) '' Metric.closedBall 0 1)) (union' : โ‹ƒ (n : โ„•), โ‹ƒ (j : cell n), โ†‘(map n j) '' Metric.closedBall 0 1 = C) (n : โ„•) (i : RelCWComplex.cell C n) :
                        Topology.CWComplex.map n i = RelCWComplex.map n i
                        source
                        ๐Ÿ“ coverage
                        theorem Topology.CWComplex.finite_mkFinite {X : Type u} [TopologicalSpace X] (C : Set X) (cell : โ„• โ†’ Type u) (map : (n : โ„•) โ†’ cell n โ†’ PartialEquiv (Fin n โ†’ โ„) X) (eventually_isEmpty_cell : โˆ€แถ  (n : โ„•) in Filter.atTop, IsEmpty (cell n)) (finite_cell : โˆ€ (n : โ„•), _root_.Finite (cell n)) (source_eq : โˆ€ (n : โ„•) (i : cell n), (map n i).source = Metric.ball 0 1) (continuousOn : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i)) (Metric.closedBall 0 1)) (continuousOn_symm : โˆ€ (n : โ„•) (i : cell n), ContinuousOn (โ†‘(map n i).symm) (map n i).target) (pairwiseDisjoint' : Set.univ.PairwiseDisjoint fun (ni : (n : โ„•) ร— cell n) => โ†‘(map ni.fst ni.snd) '' Metric.ball 0 1) (mapsTo : โˆ€ (n : โ„•) (i : cell n), Set.MapsTo (โ†‘(map n i)) (Metric.sphere 0 1) (โ‹ƒ (m : โ„•), โ‹ƒ (_ : m < n), โ‹ƒ (j : cell m), โ†‘(map m j) '' Metric.closedBall 0 1)) (union' : โ‹ƒ (n : โ„•), โ‹ƒ (j : cell n), โ†‘(map n j) '' Metric.closedBall 0 1 = C) :
                        Finite C

                        A CW complex that was constructed using CWComplex.mkFinite is finite.

                        source
                        ๐Ÿ“ coverage
                        theorem Topology.RelCWComplex.finite_of_finite_cells {X : Type u_2} [TopologicalSpace X] {C D : Set X} [RelCWComplex C D] (finite : _root_.Finite ((n : โ„•) ร— cell C n)) :
                        Finite C

                        If the collection of all cells (of any dimension) of a relative CW complex C is finite, then C is finite as a CW complex.

                        source
                        ๐Ÿ“ coverage
                        theorem Topology.RelCWComplex.finite_cells_of_finite {X : Type u_2} [TopologicalSpace X] {C D : Set X} [RelCWComplex C D] [finite : Finite C] :
                        _root_.Finite ((n : โ„•) ร— cell C n)

                        If C is finite as a CW complex then the collection of all cells (of any dimension) is finite.

                        source
                        ๐Ÿ“ coverage
                        theorem Topology.RelCWComplex.finite_iff_finite_cells {X : Type u_2} [TopologicalSpace X] {C D : Set X} [RelCWComplex C D] :
                        Finite C โ†” _root_.Finite ((n : โ„•) ร— cell C n)

                        A CW complex is finite iff the total number of its cells is finite.