Documentation

Mathlib.Topology.Sequences

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 #

Predicates #

Type classes #

Main results #

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} :

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) :

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) :

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} :

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) :

A set is sequentially closed if it is closed.

theorem mem_closure_iff_seq_limit {X : Type u_1} [TopologicalSpace X] [FrechetUrysohnSpace X] {s : Set X} {a : X} :
a closure s ∃ (x : X), (∀ (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 : XY} {a : X} {b : Y} :
Filter.Tendsto f (nhds a) (nhds b) ∀ (u : 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 : XProp) (a : X), (∀ (u : X), Filter.Tendsto u Filter.atTop (nhds a)Filter.Tendsto (f u) Filter.atTop (nhds (f a)))ContinuousAt f a) :

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

@[instance 100]

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

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@[instance 100]

Every Fréchet-Urysohn space is a sequential space.

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

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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 : XY} {s : Set Y} (hs : IsSeqClosed s) (hf : SeqContinuous f) :

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

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

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 : XY) :
theorem SequentialSpace.iSup {X : Type u_4} {ι : Sort u_3} {t : ιTopologicalSpace X} (h : ∀ (i : ι), SequentialSpace X) :
theorem SequentialSpace.sup {X : Type u_3} {t₁ : TopologicalSpace X} {t₂ : TopologicalSpace X} (h₁ : SequentialSpace X) (h₂ : SequentialSpace X) :

The quotient of a sequential space is a sequential space.

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

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instance Sigma.instSequentialSpace {ι : Type u_3} {X : ιType 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.

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theorem IsSeqCompact.subseq_of_frequently_in {X : Type u_1} [TopologicalSpace X] {s : Set X} (hs : IsSeqCompact s) {x : X} (hx : ∃ᶠ (n : ) in Filter.atTop, x n s) :
as, ∃ (φ : ), StrictMono φ Filter.Tendsto (x φ) Filter.atTop (nhds a)
theorem SeqCompactSpace.tendsto_subseq {X : Type u_1} [TopologicalSpace X] [SeqCompactSpace X] (x : X) :
∃ (a : X) (φ : ), 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 : X} (hs : IsCompact s) (hx : ∃ᶠ (n : ) in Filter.atTop, x n s) :
as, ∃ (φ : ), 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 : X} (hs : IsCompact s) (hx : ∀ (n : ), x n s) :
as, ∃ (φ : ), StrictMono φ Filter.Tendsto (x φ) Filter.atTop (nhds a)
theorem CompactSpace.tendsto_subseq {X : Type u_1} [TopologicalSpace X] [FirstCountableTopology X] [CompactSpace X] (x : X) :
∃ (a : X) (φ : ), 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 : XY} (f_cont : SeqContinuous f) {K : Set X} (K_cpt : IsSeqCompact 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 : XY} [SeqCompactSpace X] (f_cont : SeqContinuous 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 : X} (hu : ∃ᶠ (n : ) in Filter.atTop, u n s) (huc : CauchySeq u) :
xs, Filter.Tendsto u Filter.atTop (nhds x)
theorem IsSeqCompact.exists_tendsto {X : Type u_1} [UniformSpace X] {s : Set X} (hs : IsSeqCompact s) {u : X} (hu : ∀ (n : ), u n s) (huc : CauchySeq u) :
xs, Filter.Tendsto u Filter.atTop (nhds x)

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) :

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) :

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] :

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.