Documentation

Mathlib.Topology.Compactness.Lindelof

Lindelöf sets and Lindelöf spaces #

Main definitions #

We define the following properties for sets in a topological space:

Main results #

Implementation details #

def IsLindelof {X : Type u} [TopologicalSpace X] (s : Set X) :

A set s is Lindelöf if every nontrivial filter f with the countable intersection property that contains s, has a clusterpoint in s. The filter-free definition is given by isLindelof_iff_countable_subcover.

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    theorem IsLindelof.compl_mem_sets {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsLindelof s) {f : Filter X} [CountableInterFilter f] (hf : xs, s nhds x f) :
    s f

    The complement to a Lindelöf set belongs to a filter f with the countable intersection property if it belongs to each filter 𝓝 x ⊓ f, x ∈ s.

    theorem IsLindelof.compl_mem_sets_of_nhdsWithin {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsLindelof s) {f : Filter X} [CountableInterFilter f] (hf : xs, tnhdsWithin x s, t f) :
    s f

    The complement to a Lindelöf set belongs to a filter f with the countable intersection property if each x ∈ s has a neighborhood t within s such that tᶜ belongs to f.

    theorem IsLindelof.induction_on {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsLindelof s) {p : Set XProp} (hmono : ∀ ⦃s t : Set X⦄, s tp tp s) (hcountable_union : ∀ (S : Set (Set X)), S.Countable(∀ sS, p s)p (⋃₀ S)) (hnhds : xs, tnhdsWithin x s, p t) :
    p s

    If p : Set X → Prop is stable under restriction and union, and each point x of a Lindelöf set s has a neighborhood t within s such that p t, then p s holds.

    theorem IsLindelof.inter_right {X : Type u} [TopologicalSpace X] {s : Set X} {t : Set X} (hs : IsLindelof s) (ht : IsClosed t) :

    The intersection of a Lindelöf set and a closed set is a Lindelöf set.

    theorem IsLindelof.inter_left {X : Type u} [TopologicalSpace X] {s : Set X} {t : Set X} (ht : IsLindelof t) (hs : IsClosed s) :

    The intersection of a closed set and a Lindelöf set is a Lindelöf set.

    theorem IsLindelof.diff {X : Type u} [TopologicalSpace X] {s : Set X} {t : Set X} (hs : IsLindelof s) (ht : IsOpen t) :

    The set difference of a Lindelöf set and an open set is a Lindelöf set.

    theorem IsLindelof.of_isClosed_subset {X : Type u} [TopologicalSpace X] {s : Set X} {t : Set X} (hs : IsLindelof s) (ht : IsClosed t) (h : t s) :

    A closed subset of a Lindelöf set is a Lindelöf set.

    theorem IsLindelof.image_of_continuousOn {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : XY} (hs : IsLindelof s) (hf : ContinuousOn f s) :

    A continuous image of a Lindelöf set is a Lindelöf set.

    theorem IsLindelof.image {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : XY} (hs : IsLindelof s) (hf : Continuous f) :

    A continuous image of a Lindelöf set is a Lindelöf set within the codomain.

    theorem IsLindelof.adherence_nhdset {X : Type u} [TopologicalSpace X] {s : Set X} {t : Set X} {f : Filter X} [CountableInterFilter f] (hs : IsLindelof s) (hf₂ : f Filter.principal s) (ht₁ : IsOpen t) (ht₂ : xs, ClusterPt x fx t) :
    t f

    A filter with the countable intersection property that is finer than the principal filter on a Lindelöf set s contains any open set that contains all clusterpoints of s.

    theorem IsLindelof.elim_countable_subcover {X : Type u} [TopologicalSpace X] {s : Set X} {ι : Type v} (hs : IsLindelof s) (U : ιSet X) (hUo : ∀ (i : ι), IsOpen (U i)) (hsU : s ⋃ (i : ι), U i) :
    ∃ (r : Set ι), r.Countable s ir, U i

    For every open cover of a Lindelöf set, there exists a countable subcover.

    theorem IsLindelof.elim_nhds_subcover' {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsLindelof s) (U : (x : X) → x sSet X) (hU : ∀ (x : X) (hx : x s), U x hx nhds x) :
    ∃ (t : Set s), t.Countable s xt, U x
    theorem IsLindelof.elim_nhds_subcover {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsLindelof s) (U : XSet X) (hU : xs, U x nhds x) :
    ∃ (t : Set X), t.Countable (∀ xt, x s) s xt, U x
    theorem IsLindelof.indexed_countable_subcover {X : Type u} [TopologicalSpace X] {s : Set X} {ι : Type v} [Nonempty ι] (hs : IsLindelof s) (U : ιSet X) (hUo : ∀ (i : ι), IsOpen (U i)) (hsU : s ⋃ (i : ι), U i) :
    ∃ (f : ι), s ⋃ (n : ), U (f n)

    For every nonempty open cover of a Lindelöf set, there exists a subcover indexed by ℕ.

    theorem IsLindelof.disjoint_nhdsSet_left {X : Type u} [TopologicalSpace X] {s : Set X} {l : Filter X} [CountableInterFilter l] (hs : IsLindelof s) :
    Disjoint (nhdsSet s) l xs, Disjoint (nhds x) l

    The neighborhood filter of a Lindelöf set is disjoint with a filter l with the countable intersection property if and only if the neighborhood filter of each point of this set is disjoint with l.

    theorem IsLindelof.disjoint_nhdsSet_right {X : Type u} [TopologicalSpace X] {s : Set X} {l : Filter X} [CountableInterFilter l] (hs : IsLindelof s) :
    Disjoint l (nhdsSet s) xs, Disjoint l (nhds x)

    A filter l with the countable intersection property is disjoint with the neighborhood filter of a Lindelöf set if and only if it is disjoint with the neighborhood filter of each point of this set.

    theorem IsLindelof.elim_countable_subfamily_closed {X : Type u} [TopologicalSpace X] {s : Set X} {ι : Type v} (hs : IsLindelof s) (t : ιSet X) (htc : ∀ (i : ι), IsClosed (t i)) (hst : s ⋂ (i : ι), t i = ) :
    ∃ (u : Set ι), u.Countable s iu, t i =

    For every family of closed sets whose intersection avoids a Lindelö set, there exists a countable subfamily whose intersection avoids this Lindelöf set.

    theorem IsLindelof.inter_iInter_nonempty {X : Type u} [TopologicalSpace X] {s : Set X} {ι : Type v} (hs : IsLindelof s) (t : ιSet X) (htc : ∀ (i : ι), IsClosed (t i)) (hst : ∀ (u : Set ι), u.Countable (s iu, t i).Nonempty) :
    (s ⋂ (i : ι), t i).Nonempty

    To show that a Lindelöf set intersects the intersection of a family of closed sets, it is sufficient to show that it intersects every countable subfamily.

    theorem IsLindelof.elim_countable_subcover_image {X : Type u} {ι : Type u_1} [TopologicalSpace X] {s : Set X} {b : Set ι} {c : ιSet X} (hs : IsLindelof s) (hc₁ : ib, IsOpen (c i)) (hc₂ : s ib, c i) :
    b'b, b'.Countable s ib', c i

    For every open cover of a Lindelöf set, there exists a countable subcover.

    theorem isLindelof_of_countable_subcover {X : Type u} [TopologicalSpace X] {s : Set X} (h : ∀ {ι : Type u} (U : ιSet X), (∀ (i : ι), IsOpen (U i))s ⋃ (i : ι), U i∃ (t : Set ι), t.Countable s it, U i) :

    A set s is Lindelöf if for every open cover of s, there exists a countable subcover.

    theorem isLindelof_of_countable_subfamily_closed {X : Type u} [TopologicalSpace X] {s : Set X} (h : ∀ {ι : Type u} (t : ιSet X), (∀ (i : ι), IsClosed (t i))s ⋂ (i : ι), t i = ∃ (u : Set ι), u.Countable s iu, t i = ) :

    A set s is Lindelöf if for every family of closed sets whose intersection avoids s, there exists a countable subfamily whose intersection avoids s.

    theorem isLindelof_iff_countable_subcover {X : Type u} [TopologicalSpace X] {s : Set X} :
    IsLindelof s ∀ {ι : Type u} (U : ιSet X), (∀ (i : ι), IsOpen (U i))s ⋃ (i : ι), U i∃ (t : Set ι), t.Countable s it, U i

    A set s is Lindelöf if and only if for every open cover of s, there exists a countable subcover.

    theorem isLindelof_iff_countable_subfamily_closed {X : Type u} [TopologicalSpace X] {s : Set X} :
    IsLindelof s ∀ {ι : Type u} (t : ιSet X), (∀ (i : ι), IsClosed (t i))s ⋂ (i : ι), t i = ∃ (u : Set ι), u.Countable s iu, t i =

    A set s is Lindelöf if and only if for every family of closed sets whose intersection avoids s, there exists a countable subfamily whose intersection avoids s.

    @[simp]

    The empty set is a Lindelof set.

    @[simp]
    theorem isLindelof_singleton {X : Type u} [TopologicalSpace X] {x : X} :

    A singleton set is a Lindelof set.

    theorem Set.Subsingleton.isLindelof {X : Type u} [TopologicalSpace X] {s : Set X} (hs : s.Subsingleton) :
    theorem Set.Countable.isLindelof_biUnion {X : Type u} {ι : Type u_1} [TopologicalSpace X] {s : Set ι} {f : ιSet X} (hs : s.Countable) (hf : is, IsLindelof (f i)) :
    IsLindelof (⋃ is, f i)
    theorem Set.Finite.isLindelof_biUnion {X : Type u} {ι : Type u_1} [TopologicalSpace X] {s : Set ι} {f : ιSet X} (hs : s.Finite) (hf : is, IsLindelof (f i)) :
    IsLindelof (⋃ is, f i)
    theorem Finset.isLindelof_biUnion {X : Type u} {ι : Type u_1} [TopologicalSpace X] (s : Finset ι) {f : ιSet X} (hf : is, IsLindelof (f i)) :
    IsLindelof (⋃ is, f i)
    theorem isLindelof_accumulate {X : Type u} [TopologicalSpace X] {K : Set X} (hK : ∀ (n : ), IsLindelof (K n)) (n : ) :
    theorem Set.Countable.isLindelof_sUnion {X : Type u} [TopologicalSpace X] {S : Set (Set X)} (hf : S.Countable) (hc : sS, IsLindelof s) :
    theorem Set.Finite.isLindelof_sUnion {X : Type u} [TopologicalSpace X] {S : Set (Set X)} (hf : S.Finite) (hc : sS, IsLindelof s) :
    theorem isLindelof_iUnion {X : Type u} [TopologicalSpace X] {ι : Sort u_2} {f : ιSet X} [Countable ι] (h : ∀ (i : ι), IsLindelof (f i)) :
    IsLindelof (⋃ (i : ι), f i)
    theorem Set.Countable.isLindelof {X : Type u} [TopologicalSpace X] {s : Set X} (hs : s.Countable) :
    theorem Set.Finite.isLindelof {X : Type u} [TopologicalSpace X] {s : Set X} (hs : s.Finite) :
    theorem IsLindelof.countable_of_discrete {X : Type u} [TopologicalSpace X] {s : Set X} [DiscreteTopology X] (hs : IsLindelof s) :
    s.Countable
    theorem IsLindelof.union {X : Type u} [TopologicalSpace X] {s : Set X} {t : Set X} (hs : IsLindelof s) (ht : IsLindelof t) :
    theorem IsLindelof.insert {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsLindelof s) (a : X) :
    theorem isLindelof_open_iff_eq_countable_iUnion_of_isTopologicalBasis {X : Type u} {ι : Type u_1} [TopologicalSpace X] (b : ιSet X) (hb : TopologicalSpace.IsTopologicalBasis (Set.range b)) (hb' : ∀ (i : ι), IsLindelof (b i)) (U : Set X) :
    IsLindelof U IsOpen U ∃ (s : Set ι), s.Countable U = is, b i

    If X has a basis consisting of compact opens, then an open set in X is compact open iff it is a finite union of some elements in the basis

    Filter.coLindelof is the filter generated by complements to Lindelöf sets.

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      theorem hasBasis_coLindelof {X : Type u} [TopologicalSpace X] :
      (Filter.coLindelof X).HasBasis IsLindelof compl
      theorem mem_coLindelof {X : Type u} [TopologicalSpace X] {s : Set X} :
      theorem mem_coLindelof' {X : Type u} [TopologicalSpace X] {s : Set X} :

      Filter.coclosedLindelof is the filter generated by complements to closed Lindelof sets.

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        theorem hasBasis_coclosedLindelof {X : Type u} [TopologicalSpace X] :
        (Filter.coclosedLindelof X).HasBasis (fun (s : Set X) => IsClosed s IsLindelof s) compl

        X is a Lindelöf space iff every open cover has a countable subcover.

        Instances
          theorem LindelofSpace.isLindelof_univ {X : Type u_2} :
          ∀ {inst : TopologicalSpace X} [self : LindelofSpace X], IsLindelof Set.univ

          In a Lindelöf space, Set.univ is a Lindelöf set.

          @[instance 10]
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          theorem cluster_point_of_Lindelof {X : Type u} [TopologicalSpace X] [LindelofSpace X] (f : Filter X) [f.NeBot] [CountableInterFilter f] :
          ∃ (x : X), ClusterPt x f
          theorem LindelofSpace.elim_nhds_subcover {X : Type u} [TopologicalSpace X] [LindelofSpace X] (U : XSet X) (hU : ∀ (x : X), U x nhds x) :
          ∃ (t : Set X), t.Countable xt, U x = Set.univ
          theorem lindelofSpace_of_countable_subfamily_closed {X : Type u} [TopologicalSpace X] (h : ∀ {ι : Type u} (t : ιSet X), (∀ (i : ι), IsClosed (t i))⋂ (i : ι), t i = ∃ (u : Set ι), u.Countable iu, t i = ) :
          theorem IsCompact.isLindelof {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsCompact s) :

          A compact set s is Lindelöf.

          A σ-compact set s is Lindelöf

          @[instance 100]

          A compact space X is Lindelöf.

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          @[instance 100]

          A sigma-compact space X is Lindelöf.

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          X is a non-Lindelöf topological space if it is not a Lindelöf space.

          Instances
            theorem NonLindelofSpace.nonLindelof_univ {X : Type u_2} :
            ∀ {inst : TopologicalSpace X} [self : NonLindelofSpace X], ¬IsLindelof Set.univ

            In a non-Lindelöf space, Set.univ is not a Lindelöf set.

            theorem IsLindelof.ne_univ {X : Type u} [TopologicalSpace X] {s : Set X} [NonLindelofSpace X] (hs : IsLindelof s) :
            s Set.univ
            @[instance 100]

            A compact space X is Lindelöf.

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            theorem countable_cover_nhds_interior {X : Type u} [TopologicalSpace X] [LindelofSpace X] {U : XSet X} (hU : ∀ (x : X), U x nhds x) :
            ∃ (t : Set X), t.Countable xt, interior (U x) = Set.univ
            theorem countable_cover_nhds {X : Type u} [TopologicalSpace X] [LindelofSpace X] {U : XSet X} (hU : ∀ (x : X), U x nhds x) :
            ∃ (t : Set X), t.Countable xt, U x = Set.univ

            The comap of the coLindelöf filter on Y by a continuous function f : X → Y is less than or equal to the coLindelöf filter on X. This is a reformulation of the fact that images of Lindelöf sets are Lindelöf.

            theorem isLindelof_range {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] [LindelofSpace X] {f : XY} (hf : Continuous f) :
            theorem Inducing.isLindelof_iff {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : XY} (hf : Inducing f) :

            If f : X → Y is an Inducing map, the image f '' s of a set s is Lindelöf if and only if s is compact.

            theorem Embedding.isLindelof_iff {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {s : Set X} {f : XY} (hf : Embedding f) :

            If f : X → Y is an Embedding, the image f '' s of a set s is Lindelöf if and only if s is Lindelöf.

            theorem Inducing.isLindelof_preimage {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : XY} (hf : Inducing f) (hf' : IsClosed (Set.range f)) {K : Set Y} (hK : IsLindelof K) :

            The preimage of a Lindelöf set under an inducing map is a Lindelöf set.

            theorem IsClosedEmbedding.isLindelof_preimage {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : XY} (hf : IsClosedEmbedding f) {K : Set Y} (hK : IsLindelof K) :

            The preimage of a Lindelöf set under a closed embedding is a Lindelöf set.

            @[deprecated IsClosedEmbedding.isLindelof_preimage]
            theorem ClosedEmbedding.isLindelof_preimage {X : Type u} {Y : Type v} [TopologicalSpace X] [TopologicalSpace Y] {f : XY} (hf : IsClosedEmbedding f) {K : Set Y} (hK : IsLindelof K) :

            Alias of IsClosedEmbedding.isLindelof_preimage.


            The preimage of a Lindelöf set under a closed embedding is a Lindelöf set.

            A closed embedding is proper, ie, inverse images of Lindelöf sets are contained in Lindelöf. Moreover, the preimage of a Lindelöf set is Lindelöf, see IsClosedEmbedding.isLindelof_preimage.

            @[deprecated IsClosedEmbedding.tendsto_coLindelof]

            Alias of IsClosedEmbedding.tendsto_coLindelof.


            A closed embedding is proper, ie, inverse images of Lindelöf sets are contained in Lindelöf. Moreover, the preimage of a Lindelöf set is Lindelöf, see IsClosedEmbedding.isLindelof_preimage.

            theorem Subtype.isLindelof_iff {X : Type u} [TopologicalSpace X] {p : XProp} {s : Set { x : X // p x }} :
            IsLindelof s IsLindelof (Subtype.val '' s)

            Sets of subtype are Lindelöf iff the image under a coercion is.

            theorem IsLindelof.countable {X : Type u} [TopologicalSpace X] {s : Set X} (hs : IsLindelof s) (hs' : DiscreteTopology s) :
            s.Countable
            @[deprecated IsClosedEmbedding.nonLindelofSpace]

            Alias of IsClosedEmbedding.nonLindelofSpace.

            @[deprecated IsClosedEmbedding.LindelofSpace]

            Alias of IsClosedEmbedding.LindelofSpace.

            @[instance 100]

            Countable topological spaces are Lindelof.

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            The disjoint union of two Lindelöf spaces is Lindelöf.

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            instance instLindelofSpaceSigmaOfCountable {ι : Type u_1} {X : ιType u_2} [Countable ι] [(i : ι) → TopologicalSpace (X i)] [∀ (i : ι), LindelofSpace (X i)] :
            LindelofSpace ((i : ι) × X i)
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            instance Quot.LindelofSpace {X : Type u} [TopologicalSpace X] {r : XXProp} [LindelofSpace X] :
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            A continuous image of a Lindelöf set is a Lindelöf set within the codomain.

            A set s is Hereditarily Lindelöf if every subset is a Lindelof set. We require this only for open sets in the definition, and then conclude that this holds for all sets by ADD.

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              Type class for Hereditarily Lindelöf spaces.

              Instances

                In a Hereditarily Lindelöf space, Set.univ is a Hereditarily Lindelöf set.

                @[instance 100]
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                theorem eq_open_union_countable {X : Type u} [TopologicalSpace X] [HereditarilyLindelofSpace X] {ι : Type u} (U : ιSet X) (h : ∀ (i : ι), IsOpen (U i)) :
                ∃ (t : Set ι), t.Countable it, U i = ⋃ (i : ι), U i
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