Sequences in topological spaces

In this file we prove theorems about relations between closure/compactness/continuity etc and their sequential counterparts.

Main definitions

The following notions are defined in Topology/Defs/Sequences. We build theory about these definitions here, so we remind the definitions.

Set operation

• seqClosure s: sequential closure of a set, the set of limits of sequences of points of s;

Predicates

• IsSeqClosed s: predicate saying that a set is sequentially closed, i.e., seqClosure s ⊆ s;
• SeqContinuous f: predicate saying that a function is sequentially continuous, i.e., for any sequence u : ℕ → X that converges to a point x, the sequence f ∘ u converges to f x;
• IsSeqCompact s: predicate saying that a set is sequentially compact, i.e., every sequence taking values in s has a converging subsequence.

Type classes

• FrechetUrysohnSpace X: a typeclass saying that a topological space is a Fréchet-Urysohn space, i.e., the sequential closure of any set is equal to its closure.
• SequentialSpace X: a typeclass saying that a topological space is a sequential space, i.e., any sequentially closed set in this space is closed. This condition is weaker than being a Fréchet-Urysohn space.
• SeqCompactSpace X: a typeclass saying that a topological space is sequentially compact, i.e., every sequence in X has a converging subsequence.

Main results

• seqClosure_subset_closure: closure of a set includes its sequential closure;
• IsClosed.isSeqClosed: a closed set is sequentially closed;
• IsSeqClosed.seqClosure_eq: sequential closure of a sequentially closed set s is equal to s;
• seqClosure_eq_closure: in a Fréchet-Urysohn space, the sequential closure of a set is equal to its closure;
• tendsto_nhds_iff_seq_tendsto, FrechetUrysohnSpace.of_seq_tendsto_imp_tendsto: a topological space is a Fréchet-Urysohn space if and only if sequential convergence implies convergence;
• FirstCountableTopology.frechetUrysohnSpace: every topological space with first countable topology is a Fréchet-Urysohn space;
• FrechetUrysohnSpace.to_sequentialSpace: every Fréchet-Urysohn space is a sequential space;
• IsSeqCompact.isCompact: a sequentially compact set in a uniform space with countably generated uniformity is compact.

Tags

sequentially closed, sequentially compact, sequential space

Sequential closures, sequential continuity, and sequential spaces.

theorem subset_seqClosure {X : Type u_1} [TopologicalSpace X] {s : Set X} : s ⊆ seqClosure s

theorem seqClosure_subset_closure {X : Type u_1} [TopologicalSpace X] {s : Set X} : seqClosure s ⊆ closure s

The sequential closure of a set is contained in the closure of that set. The converse is not true.

theorem IsSeqClosed.seqClosure_eq {X : Type u_1} [TopologicalSpace X] {s : Set X} (hs : IsSeqClosed s) : seqClosure s = s

The sequential closure of a sequentially closed set is the set itself.

theorem isSeqClosed_of_seqClosure_eq {X : Type u_1} [TopologicalSpace X] {s : Set X} (hs : seqClosure s = s) : IsSeqClosed s

If a set is equal to its sequential closure, then it is sequentially closed.

theorem isSeqClosed_iff {X : Type u_1} [TopologicalSpace X] {s : Set X} : IsSeqClosed s ↔ seqClosure s = s

A set is sequentially closed iff it is equal to its sequential closure.

theorem IsClosed.isSeqClosed {X : Type u_1} [TopologicalSpace X] {s : Set X} (hc : IsClosed s) : IsSeqClosed s

A set is sequentially closed if it is closed.

theorem seqClosure_eq_closure {X : Type u_1} [TopologicalSpace X] [FrechetUrysohnSpace X] (s : Set X) : seqClosure s = closure s

theorem mem_closure_iff_seq_limit {X : Type u_1} [TopologicalSpace X] [FrechetUrysohnSpace X] {s : Set X} {a : X} : a ∈ closure s ↔ ∃ (x : ℕ → X), (∀ (n : ℕ), x n ∈ s) ∧ Filter.Tendsto x Filter.atTop (nhds a)

In a Fréchet-Urysohn space, a point belongs to the closure of a set iff it is a limit of a sequence taking values in this set.

theorem tendsto_nhds_iff_seq_tendsto {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [FrechetUrysohnSpace X] {f : X → Y} {a : X} {b : Y} : Filter.Tendsto f (nhds a) (nhds b) ↔ ∀ (u : ℕ → X), Filter.Tendsto u Filter.atTop (nhds a) → Filter.Tendsto (f ∘ u) Filter.atTop (nhds b)

If the domain of a function f : α → β is a Fréchet-Urysohn space, then convergence is equivalent to sequential convergence. See also Filter.tendsto_iff_seq_tendsto for a version that works for any pair of filters assuming that the filter in the domain is countably generated.

This property is equivalent to the definition of FrechetUrysohnSpace, see FrechetUrysohnSpace.of_seq_tendsto_imp_tendsto.

theorem FrechetUrysohnSpace.of_seq_tendsto_imp_tendsto {X : Type u_1} [TopologicalSpace X] (h : ∀ (f : X → Prop) (a : X), (∀ (u : ℕ → X), Filter.Tendsto u Filter.atTop (nhds a) → Filter.Tendsto (f ∘ u) Filter.atTop (nhds (f a))) → ContinuousAt f a) : FrechetUrysohnSpace X

An alternative construction for FrechetUrysohnSpace: if sequential convergence implies convergence, then the space is a Fréchet-Urysohn space.

@[instance 100]

instance FirstCountableTopology.frechetUrysohnSpace {X : Type u_1} [TopologicalSpace X] [FirstCountableTopology X] : FrechetUrysohnSpace X

Every first-countable space is a Fréchet-Urysohn space.

Equations

• ⋯ = ⋯

@[instance 100]

instance FrechetUrysohnSpace.to_sequentialSpace {X : Type u_1} [TopologicalSpace X] [FrechetUrysohnSpace X] : SequentialSpace X

Every Fréchet-Urysohn space is a sequential space.

Equations

• ⋯ = ⋯

theorem Inducing.frechetUrysohnSpace {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [FrechetUrysohnSpace Y] {f : X → Y} (hf : Inducing f) : FrechetUrysohnSpace X

instance Subtype.instFrechetUrysohnSpace {X : Type u_1} [TopologicalSpace X] [FrechetUrysohnSpace X] {p : X → Prop} : FrechetUrysohnSpace (Subtype p)

Subtype of a Fréchet-Urysohn space is a Fréchet-Urysohn space.

Equations

• ⋯ = ⋯

theorem isSeqClosed_iff_isClosed {X : Type u_1} [TopologicalSpace X] [SequentialSpace X] {M : Set X} : IsSeqClosed M ↔ IsClosed M

In a sequential space, a set is closed iff it's sequentially closed.

theorem IsSeqClosed.preimage {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} {s : Set Y} (hs : IsSeqClosed s) (hf : SeqContinuous f) : IsSeqClosed (f ⁻¹' s)

The preimage of a sequentially closed set under a sequentially continuous map is sequentially closed.

theorem Continuous.seqContinuous {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (hf : Continuous f) : SeqContinuous f

theorem SeqContinuous.continuous {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [SequentialSpace X] {f : X → Y} (hf : SeqContinuous f) : Continuous f

A sequentially continuous function defined on a sequential space is continuous.

theorem continuous_iff_seqContinuous {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [SequentialSpace X] {f : X → Y} : Continuous f ↔ SeqContinuous f

If the domain of a function is a sequential space, then continuity of this function is equivalent to its sequential continuity.

theorem SequentialSpace.coinduced {X : Type u_1} [TopologicalSpace X] [SequentialSpace X] {Y : Type u_3} (f : X → Y) : SequentialSpace Y

theorem SequentialSpace.iSup {X : Type u_4} {ι : Sort u_3} {t : ι → TopologicalSpace X} (h : ∀ (i : ι), SequentialSpace X) : SequentialSpace X

theorem SequentialSpace.sup {X : Type u_3} {t₁ : TopologicalSpace X} {t₂ : TopologicalSpace X} (h₁ : SequentialSpace X) (h₂ : SequentialSpace X) : SequentialSpace X

theorem QuotientMap.sequentialSpace {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [SequentialSpace X] {f : X → Y} (hf : QuotientMap f) : SequentialSpace Y

instance Quotient.instSequentialSpace {X : Type u_1} [TopologicalSpace X] [SequentialSpace X] {s : Setoid X} : SequentialSpace (Quotient s)

The quotient of a sequential space is a sequential space.

Equations

• ⋯ = ⋯

instance Sum.instSequentialSpace {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] [SequentialSpace X] [SequentialSpace Y] : SequentialSpace (X ⊕ Y)

The sum (disjoint union) of two sequential spaces is a sequential space.

Equations

• ⋯ = ⋯

instance Sigma.instSequentialSpace {ι : Type u_3} {X : ι → Type u_4} [(i : ι) → TopologicalSpace (X i)] [∀ (i : ι), SequentialSpace (X i)] : SequentialSpace ((i : ι) × X i)

The disjoint union of an indexed family of sequential spaces is a sequential space.

Equations

• ⋯ = ⋯

theorem IsSeqCompact.subseq_of_frequently_in {X : Type u_1} [TopologicalSpace X] {s : Set X} (hs : IsSeqCompact s) {x : ℕ → X} (hx : ∃ᶠ (n : ℕ) in Filter.atTop, x n ∈ s) : ∃ a ∈ s, ∃ (φ : ℕ → ℕ), StrictMono φ ∧ Filter.Tendsto (x ∘ φ) Filter.atTop (nhds a)

theorem SeqCompactSpace.tendsto_subseq {X : Type u_1} [TopologicalSpace X] [SeqCompactSpace X] (x : ℕ → X) : ∃ (a : X) (φ : ℕ → ℕ), StrictMono φ ∧ Filter.Tendsto (x ∘ φ) Filter.atTop (nhds a)

theorem IsCompact.isSeqCompact {X : Type u_1} [TopologicalSpace X] [FirstCountableTopology X] {s : Set X} (hs : IsCompact s) : IsSeqCompact s

theorem IsCompact.tendsto_subseq' {X : Type u_1} [TopologicalSpace X] [FirstCountableTopology X] {s : Set X} {x : ℕ → X} (hs : IsCompact s) (hx : ∃ᶠ (n : ℕ) in Filter.atTop, x n ∈ s) : ∃ a ∈ s, ∃ (φ : ℕ → ℕ), StrictMono φ ∧ Filter.Tendsto (x ∘ φ) Filter.atTop (nhds a)

theorem IsCompact.tendsto_subseq {X : Type u_1} [TopologicalSpace X] [FirstCountableTopology X] {s : Set X} {x : ℕ → X} (hs : IsCompact s) (hx : ∀ (n : ℕ), x n ∈ s) : ∃ a ∈ s, ∃ (φ : ℕ → ℕ), StrictMono φ ∧ Filter.Tendsto (x ∘ φ) Filter.atTop (nhds a)

@[instance 100]

instance FirstCountableTopology.seq_compact_of_compact {X : Type u_1} [TopologicalSpace X] [FirstCountableTopology X] [CompactSpace X] : SeqCompactSpace X

Equations

• ⋯ = ⋯

theorem CompactSpace.tendsto_subseq {X : Type u_1} [TopologicalSpace X] [FirstCountableTopology X] [CompactSpace X] (x : ℕ → X) : ∃ (a : X) (φ : ℕ → ℕ), StrictMono φ ∧ Filter.Tendsto (x ∘ φ) Filter.atTop (nhds a)

theorem IsSeqCompact.image {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} (f_cont : SeqContinuous f) {K : Set X} (K_cpt : IsSeqCompact K) : IsSeqCompact (f '' K)

Sequential compactness of sets is preserved under sequentially continuous functions.

theorem IsSeqCompact.range {X : Type u_1} {Y : Type u_2} [TopologicalSpace X] [TopologicalSpace Y] {f : X → Y} [SeqCompactSpace X] (f_cont : SeqContinuous f) : IsSeqCompact (Set.range f)

The range of sequentially continuous function on a sequentially compact space is sequentially compact.

theorem IsSeqCompact.exists_tendsto_of_frequently_mem {X : Type u_1} [UniformSpace X] {s : Set X} (hs : IsSeqCompact s) {u : ℕ → X} (hu : ∃ᶠ (n : ℕ) in Filter.atTop, u n ∈ s) (huc : CauchySeq u) : ∃ x ∈ s, Filter.Tendsto u Filter.atTop (nhds x)

theorem IsSeqCompact.exists_tendsto {X : Type u_1} [UniformSpace X] {s : Set X} (hs : IsSeqCompact s) {u : ℕ → X} (hu : ∀ (n : ℕ), u n ∈ s) (huc : CauchySeq u) : ∃ x ∈ s, Filter.Tendsto u Filter.atTop (nhds x)

theorem IsSeqCompact.totallyBounded {X : Type u_1} [UniformSpace X] {s : Set X} (h : IsSeqCompact s) : TotallyBounded s

A sequentially compact set in a uniform space is totally bounded.

theorem IsSeqCompact.isComplete {X : Type u_1} [UniformSpace X] {s : Set X} [(uniformity X).IsCountablyGenerated] (hs : IsSeqCompact s) : IsComplete s

A sequentially compact set in a uniform set with countably generated uniformity filter is complete.

theorem IsSeqCompact.isCompact {X : Type u_1} [UniformSpace X] {s : Set X} [(uniformity X).IsCountablyGenerated] (hs : IsSeqCompact s) : IsCompact s

If 𝓤 β is countably generated, then any sequentially compact set is compact.

theorem UniformSpace.isCompact_iff_isSeqCompact {X : Type u_1} [UniformSpace X] {s : Set X} [(uniformity X).IsCountablyGenerated] : IsCompact s ↔ IsSeqCompact s

A version of Bolzano-Weierstrass: in a uniform space with countably generated uniformity filter (e.g., in a metric space), a set is compact if and only if it is sequentially compact.

theorem UniformSpace.compactSpace_iff_seqCompactSpace {X : Type u_1} [UniformSpace X] [(uniformity X).IsCountablyGenerated] : CompactSpace X ↔ SeqCompactSpace X