fact stringlengths 6 3.84k | type stringclasses 11 values | library stringclasses 32 values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
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cauchy_prod_iff [UniformSpace β] {F : Filter (α × β)} :
Cauchy F ↔ Cauchy (map Prod.fst F) ∧ Cauchy (map Prod.snd F) := by
simp_rw [instUniformSpaceProd, ← cauchy_comap_uniformSpace, ← cauchy_inf_uniformSpace] | lemma | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchy_prod_iff | null |
Cauchy.prod [UniformSpace β] {f : Filter α} {g : Filter β} (hf : Cauchy f) (hg : Cauchy g) :
Cauchy (f ×ˢ g) := by
have := hf.1; have := hg.1
simpa [cauchy_prod_iff, hf.1] using ⟨hf, hg⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | Cauchy.prod | null |
le_nhds_of_cauchy_adhp_aux {f : Filter α} {x : α}
(adhs : ∀ s ∈ 𝓤 α, ∃ t ∈ f, t ×ˢ t ⊆ s ∧ ∃ y, (x, y) ∈ s ∧ y ∈ t) : f ≤ 𝓝 x := by
intro s hs
rcases comp_mem_uniformity_sets (mem_nhds_uniformity_iff_right.1 hs) with ⟨U, U_mem, hU⟩
rcases adhs U U_mem with ⟨t, t_mem, ht, y, hxy, hy⟩
apply mem_of_superset t_mem
exact fun z hz => hU (prodMk_mem_compRel hxy (ht <| mk_mem_prod hy hz)) rfl | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | le_nhds_of_cauchy_adhp_aux | The common part of the proofs of `le_nhds_of_cauchy_adhp` and
`SequentiallyComplete.le_nhds_of_seq_tendsto_nhds`: if for any entourage `s`
one can choose a set `t ∈ f` of diameter `s` such that it contains a point `y`
with `(x, y) ∈ s`, then `f` converges to `x`. |
le_nhds_of_cauchy_adhp {f : Filter α} {x : α} (hf : Cauchy f) (adhs : ClusterPt x f) :
f ≤ 𝓝 x :=
le_nhds_of_cauchy_adhp_aux
(fun s hs => by
obtain ⟨t, t_mem, ht⟩ : ∃ t ∈ f, t ×ˢ t ⊆ s := (cauchy_iff.1 hf).2 s hs
use t, t_mem, ht
exact forall_mem_nonempty_iff_neBot.2 adhs _ (inter_mem_inf (mem_nhds_left x hs) t_mem)) | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | le_nhds_of_cauchy_adhp | If `x` is an adherent (cluster) point for a Cauchy filter `f`, then it is a limit point
for `f`. |
le_nhds_iff_adhp_of_cauchy {f : Filter α} {x : α} (hf : Cauchy f) :
f ≤ 𝓝 x ↔ ClusterPt x f :=
⟨fun h => ClusterPt.of_le_nhds' h hf.1, le_nhds_of_cauchy_adhp hf⟩
nonrec theorem Cauchy.map [UniformSpace β] {f : Filter α} {m : α → β} (hf : Cauchy f)
(hm : UniformContinuous m) : Cauchy (map m f) :=
⟨hf.1.map _,
calc
map m f ×ˢ map m f = map (Prod.map m m) (f ×ˢ f) := Filter.prod_map_map_eq
_ ≤ Filter.map (Prod.map m m) (𝓤 α) := map_mono hf.right
_ ≤ 𝓤 β := hm⟩
nonrec theorem Cauchy.comap [UniformSpace β] {f : Filter β} {m : α → β} (hf : Cauchy f)
(hm : comap (fun p : α × α => (m p.1, m p.2)) (𝓤 β) ≤ 𝓤 α) [NeBot (comap m f)] :
Cauchy (comap m f) :=
⟨‹_›,
calc
comap m f ×ˢ comap m f = comap (Prod.map m m) (f ×ˢ f) := prod_comap_comap_eq
_ ≤ comap (Prod.map m m) (𝓤 β) := comap_mono hf.right
_ ≤ 𝓤 α := hm⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | le_nhds_iff_adhp_of_cauchy | null |
Cauchy.comap' [UniformSpace β] {f : Filter β} {m : α → β} (hf : Cauchy f)
(hm : Filter.comap (fun p : α × α => (m p.1, m p.2)) (𝓤 β) ≤ 𝓤 α)
(_ : NeBot (Filter.comap m f)) : Cauchy (Filter.comap m f) :=
hf.comap hm | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | Cauchy.comap' | null |
CauchySeq [Preorder β] (u : β → α) :=
Cauchy (atTop.map u) | def | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | CauchySeq | Cauchy sequences. Usually defined on ℕ, but often it is also useful to say that a function
defined on ℝ is Cauchy at +∞ to deduce convergence. Therefore, we define it in a type class that
is general enough to cover both ℕ and ℝ, which are the main motivating examples. |
CauchySeq.tendsto_uniformity [Preorder β] {u : β → α} (h : CauchySeq u) :
Tendsto (Prod.map u u) atTop (𝓤 α) := by
simpa only [Tendsto, prod_map_map_eq', prod_atTop_atTop_eq] using h.right | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | CauchySeq.tendsto_uniformity | null |
CauchySeq.nonempty [Preorder β] {u : β → α} (hu : CauchySeq u) : Nonempty β :=
@nonempty_of_neBot _ _ <| (map_neBot_iff _).1 hu.1 | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | CauchySeq.nonempty | null |
CauchySeq.mem_entourage {β : Type*} [SemilatticeSup β] {u : β → α} (h : CauchySeq u)
{V : Set (α × α)} (hV : V ∈ 𝓤 α) : ∃ k₀, ∀ i j, k₀ ≤ i → k₀ ≤ j → (u i, u j) ∈ V := by
haveI := h.nonempty
have := h.tendsto_uniformity; rw [← prod_atTop_atTop_eq] at this
simpa [MapsTo] using atTop_basis.prod_self.tendsto_left_iff.1 this V hV | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | CauchySeq.mem_entourage | null |
Filter.Tendsto.cauchySeq [SemilatticeSup β] [Nonempty β] {f : β → α} {x}
(hx : Tendsto f atTop (𝓝 x)) : CauchySeq f :=
hx.cauchy_map | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | Filter.Tendsto.cauchySeq | null |
cauchySeq_const [SemilatticeSup β] [Nonempty β] (x : α) : CauchySeq fun _ : β => x :=
tendsto_const_nhds.cauchySeq | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchySeq_const | null |
cauchySeq_iff_tendsto [Nonempty β] [SemilatticeSup β] {u : β → α} :
CauchySeq u ↔ Tendsto (Prod.map u u) atTop (𝓤 α) :=
cauchy_map_iff'.trans <| by simp only [prod_atTop_atTop_eq, Prod.map_def] | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchySeq_iff_tendsto | null |
CauchySeq.comp_tendsto {γ} [Preorder β] [SemilatticeSup γ] [Nonempty γ] {f : β → α}
(hf : CauchySeq f) {g : γ → β} (hg : Tendsto g atTop atTop) : CauchySeq (f ∘ g) :=
⟨inferInstance, le_trans (prod_le_prod.mpr ⟨Tendsto.comp le_rfl hg, Tendsto.comp le_rfl hg⟩) hf.2⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | CauchySeq.comp_tendsto | null |
CauchySeq.comp_injective [SemilatticeSup β] [NoMaxOrder β] [Nonempty β] {u : ℕ → α}
(hu : CauchySeq u) {f : β → ℕ} (hf : Injective f) : CauchySeq (u ∘ f) :=
hu.comp_tendsto <| Nat.cofinite_eq_atTop ▸ hf.tendsto_cofinite.mono_left atTop_le_cofinite | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | CauchySeq.comp_injective | null |
Function.Bijective.cauchySeq_comp_iff {f : ℕ → ℕ} (hf : Bijective f) (u : ℕ → α) :
CauchySeq (u ∘ f) ↔ CauchySeq u := by
refine ⟨fun H => ?_, fun H => H.comp_injective hf.injective⟩
lift f to ℕ ≃ ℕ using hf
simpa only [Function.comp_def, f.apply_symm_apply] using H.comp_injective f.symm.injective | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | Function.Bijective.cauchySeq_comp_iff | null |
CauchySeq.subseq_subseq_mem {V : ℕ → Set (α × α)} (hV : ∀ n, V n ∈ 𝓤 α) {u : ℕ → α}
(hu : CauchySeq u) {f g : ℕ → ℕ} (hf : Tendsto f atTop atTop) (hg : Tendsto g atTop atTop) :
∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n, ((u ∘ f ∘ φ) n, (u ∘ g ∘ φ) n) ∈ V n := by
rw [cauchySeq_iff_tendsto] at hu
exact ((hu.comp <| hf.prod_atTop hg).comp tendsto_atTop_diagonal).subseq_mem hV | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | CauchySeq.subseq_subseq_mem | null |
cauchySeq_iff' {u : ℕ → α} :
CauchySeq u ↔ ∀ V ∈ 𝓤 α, ∀ᶠ k in atTop, k ∈ Prod.map u u ⁻¹' V :=
cauchySeq_iff_tendsto | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchySeq_iff' | null |
cauchySeq_iff {u : ℕ → α} :
CauchySeq u ↔ ∀ V ∈ 𝓤 α, ∃ N, ∀ k ≥ N, ∀ l ≥ N, (u k, u l) ∈ V := by
simp only [cauchySeq_iff', Filter.eventually_atTop_prod_self', mem_preimage, Prod.map_apply] | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchySeq_iff | null |
CauchySeq.prodMap {γ δ} [UniformSpace β] [Preorder γ] [Preorder δ] {u : γ → α} {v : δ → β}
(hu : CauchySeq u) (hv : CauchySeq v) : CauchySeq (Prod.map u v) := by
simpa only [CauchySeq, prod_map_map_eq', prod_atTop_atTop_eq] using hu.prod hv
@[deprecated (since := "2025-03-10")]
alias CauchySeq.prod_map := CauchySeq.prodMap | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | CauchySeq.prodMap | null |
CauchySeq.prodMk {γ} [UniformSpace β] [Preorder γ] {u : γ → α} {v : γ → β}
(hu : CauchySeq u) (hv : CauchySeq v) : CauchySeq fun x => (u x, v x) :=
haveI := hu.1.of_map
(Cauchy.prod hu hv).mono (tendsto_map.prodMk tendsto_map)
@[deprecated (since := "2025-03-10")]
alias CauchySeq.prod := CauchySeq.prodMk | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | CauchySeq.prodMk | null |
CauchySeq.eventually_eventually [Preorder β] {u : β → α} (hu : CauchySeq u)
{V : Set (α × α)} (hV : V ∈ 𝓤 α) : ∀ᶠ k in atTop, ∀ᶠ l in atTop, (u k, u l) ∈ V :=
eventually_atTop_curry <| hu.tendsto_uniformity hV | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | CauchySeq.eventually_eventually | null |
UniformContinuous.comp_cauchySeq {γ} [UniformSpace β] [Preorder γ] {f : α → β}
(hf : UniformContinuous f) {u : γ → α} (hu : CauchySeq u) : CauchySeq (f ∘ u) :=
hu.map hf | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | UniformContinuous.comp_cauchySeq | null |
CauchySeq.subseq_mem {V : ℕ → Set (α × α)} (hV : ∀ n, V n ∈ 𝓤 α) {u : ℕ → α}
(hu : CauchySeq u) : ∃ φ : ℕ → ℕ, StrictMono φ ∧ ∀ n, (u <| φ (n + 1), u <| φ n) ∈ V n := by
have : ∀ n, ∃ N, ∀ k ≥ N, ∀ l ≥ k, (u l, u k) ∈ V n := fun n => by
rw [cauchySeq_iff] at hu
rcases hu _ (hV n) with ⟨N, H⟩
exact ⟨N, fun k hk l hl => H _ (le_trans hk hl) _ hk⟩
obtain ⟨φ : ℕ → ℕ, φ_extr : StrictMono φ, hφ : ∀ n, ∀ l ≥ φ n, (u l, u <| φ n) ∈ V n⟩ :=
extraction_forall_of_eventually' this
exact ⟨φ, φ_extr, fun n => hφ _ _ (φ_extr <| Nat.lt_add_one n).le⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | CauchySeq.subseq_mem | null |
Filter.Tendsto.subseq_mem_entourage {V : ℕ → Set (α × α)} (hV : ∀ n, V n ∈ 𝓤 α) {u : ℕ → α}
{a : α} (hu : Tendsto u atTop (𝓝 a)) : ∃ φ : ℕ → ℕ, StrictMono φ ∧ (u (φ 0), a) ∈ V 0 ∧
∀ n, (u <| φ (n + 1), u <| φ n) ∈ V (n + 1) := by
rcases mem_atTop_sets.1 (hu (ball_mem_nhds a (symm_le_uniformity <| hV 0))) with ⟨n, hn⟩
rcases (hu.comp (tendsto_add_atTop_nat n)).cauchySeq.subseq_mem fun n => hV (n + 1) with
⟨φ, φ_mono, hφV⟩
exact ⟨fun k => φ k + n, φ_mono.add_const _, hn _ le_add_self, hφV⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | Filter.Tendsto.subseq_mem_entourage | null |
tendsto_nhds_of_cauchySeq_of_subseq [Preorder β] {u : β → α} (hu : CauchySeq u)
{ι : Type*} {f : ι → β} {p : Filter ι} [NeBot p] (hf : Tendsto f p atTop) {a : α}
(ha : Tendsto (u ∘ f) p (𝓝 a)) : Tendsto u atTop (𝓝 a) :=
le_nhds_of_cauchy_adhp hu (ha.mapClusterPt.of_comp hf) | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | tendsto_nhds_of_cauchySeq_of_subseq | If a Cauchy sequence has a convergent subsequence, then it converges. |
cauchySeq_shift {u : ℕ → α} (k : ℕ) : CauchySeq (fun n ↦ u (n + k)) ↔ CauchySeq u := by
constructor <;> intro h
· rw [cauchySeq_iff] at h ⊢
intro V mV
obtain ⟨N, h⟩ := h V mV
use N + k
intro a ha b hb
convert h (a - k) (Nat.le_sub_of_add_le ha) (b - k) (Nat.le_sub_of_add_le hb) <;> omega
· exact h.comp_tendsto (tendsto_add_atTop_nat k) | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchySeq_shift | Any shift of a Cauchy sequence is also a Cauchy sequence. |
Filter.HasBasis.cauchySeq_iff {γ} [Nonempty β] [SemilatticeSup β] {u : β → α} {p : γ → Prop}
{s : γ → Set (α × α)} (h : (𝓤 α).HasBasis p s) :
CauchySeq u ↔ ∀ i, p i → ∃ N, ∀ m, N ≤ m → ∀ n, N ≤ n → (u m, u n) ∈ s i := by
rw [cauchySeq_iff_tendsto, ← prod_atTop_atTop_eq]
refine (atTop_basis.prod_self.tendsto_iff h).trans ?_
simp only [true_and, Prod.forall, mem_prod_eq,
mem_Ici, and_imp, Prod.map, @forall_swap (_ ≤ _) β] | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | Filter.HasBasis.cauchySeq_iff | null |
Filter.HasBasis.cauchySeq_iff' {γ} [Nonempty β] [SemilatticeSup β] {u : β → α}
{p : γ → Prop} {s : γ → Set (α × α)} (H : (𝓤 α).HasBasis p s) :
CauchySeq u ↔ ∀ i, p i → ∃ N, ∀ n ≥ N, (u n, u N) ∈ s i := by
refine H.cauchySeq_iff.trans ⟨fun h i hi => ?_, fun h i hi => ?_⟩
· exact (h i hi).imp fun N hN n hn => hN n hn N le_rfl
· rcases comp_symm_of_uniformity (H.mem_of_mem hi) with ⟨t, ht, ht', hts⟩
rcases H.mem_iff.1 ht with ⟨j, hj, hjt⟩
refine (h j hj).imp fun N hN m hm n hn => hts ⟨u N, hjt ?_, ht' <| hjt ?_⟩
exacts [hN m hm, hN n hn] | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | Filter.HasBasis.cauchySeq_iff' | null |
cauchySeq_of_controlled [SemilatticeSup β] [Nonempty β] (U : β → Set (α × α))
(hU : ∀ s ∈ 𝓤 α, ∃ n, U n ⊆ s) {f : β → α}
(hf : ∀ ⦃N m n : β⦄, N ≤ m → N ≤ n → (f m, f n) ∈ U N) : CauchySeq f :=
cauchySeq_iff_tendsto.2
(by
intro s hs
rw [mem_map, mem_atTop_sets]
obtain ⟨N, hN⟩ := hU s hs
refine ⟨(N, N), fun mn hmn => ?_⟩
obtain ⟨m, n⟩ := mn
exact hN (hf hmn.1 hmn.2)) | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchySeq_of_controlled | null |
isComplete_iff_clusterPt {s : Set α} :
IsComplete s ↔ ∀ l, Cauchy l → l ≤ 𝓟 s → ∃ x ∈ s, ClusterPt x l :=
forall₃_congr fun _ hl _ => exists_congr fun _ => and_congr_right fun _ =>
le_nhds_iff_adhp_of_cauchy hl | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | isComplete_iff_clusterPt | null |
isComplete_iff_ultrafilter {s : Set α} :
IsComplete s ↔ ∀ l : Ultrafilter α, Cauchy (l : Filter α) → ↑l ≤ 𝓟 s → ∃ x ∈ s, ↑l ≤ 𝓝 x := by
refine ⟨fun h l => h l, fun H => isComplete_iff_clusterPt.2 fun l hl hls => ?_⟩
haveI := hl.1
rcases H (Ultrafilter.of l) hl.ultrafilter_of ((Ultrafilter.of_le l).trans hls) with ⟨x, hxs, hxl⟩
exact ⟨x, hxs, (ClusterPt.of_le_nhds hxl).mono (Ultrafilter.of_le l)⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | isComplete_iff_ultrafilter | null |
isComplete_iff_ultrafilter' {s : Set α} :
IsComplete s ↔ ∀ l : Ultrafilter α, Cauchy (l : Filter α) → s ∈ l → ∃ x ∈ s, ↑l ≤ 𝓝 x :=
isComplete_iff_ultrafilter.trans <| by simp only [le_principal_iff, Ultrafilter.mem_coe] | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | isComplete_iff_ultrafilter' | null |
protected IsComplete.union {s t : Set α} (hs : IsComplete s) (ht : IsComplete t) :
IsComplete (s ∪ t) := by
simp only [isComplete_iff_ultrafilter', Ultrafilter.union_mem_iff, or_imp] at *
exact fun l hl =>
⟨fun hsl => (hs l hl hsl).imp fun x hx => ⟨Or.inl hx.1, hx.2⟩, fun htl =>
(ht l hl htl).imp fun x hx => ⟨Or.inr hx.1, hx.2⟩⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | IsComplete.union | null |
isComplete_iUnion_separated {ι : Sort*} {s : ι → Set α} (hs : ∀ i, IsComplete (s i))
{U : Set (α × α)} (hU : U ∈ 𝓤 α) (hd : ∀ (i j : ι), ∀ x ∈ s i, ∀ y ∈ s j, (x, y) ∈ U → i = j) :
IsComplete (⋃ i, s i) := by
set S := ⋃ i, s i
intro l hl hls
rw [le_principal_iff] at hls
obtain ⟨hl_ne, hl'⟩ := cauchy_iff.1 hl
obtain ⟨t, htS, htl, htU⟩ : ∃ t, t ⊆ S ∧ t ∈ l ∧ t ×ˢ t ⊆ U := by
rcases hl' U hU with ⟨t, htl, htU⟩
refine ⟨t ∩ S, inter_subset_right, inter_mem htl hls, Subset.trans ?_ htU⟩
gcongr <;> apply inter_subset_left
obtain ⟨i, hi⟩ : ∃ i, t ⊆ s i := by
rcases Filter.nonempty_of_mem htl with ⟨x, hx⟩
rcases mem_iUnion.1 (htS hx) with ⟨i, hi⟩
refine ⟨i, fun y hy => ?_⟩
rcases mem_iUnion.1 (htS hy) with ⟨j, hj⟩
rwa [hd i j x hi y hj (htU <| mk_mem_prod hx hy)]
rcases hs i l hl (le_principal_iff.2 <| mem_of_superset htl hi) with ⟨x, hxs, hlx⟩
exact ⟨x, mem_iUnion.2 ⟨i, hxs⟩, hlx⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | isComplete_iUnion_separated | null |
CompleteSpace (α : Type u) [UniformSpace α] : Prop where
/-- In a complete uniform space, every Cauchy filter converges. -/
complete : ∀ {f : Filter α}, Cauchy f → ∃ x, f ≤ 𝓝 x | class | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | CompleteSpace | A complete space is defined here using uniformities. A uniform space
is complete if every Cauchy filter converges. |
complete_univ {α : Type u} [UniformSpace α] [CompleteSpace α] :
IsComplete (univ : Set α) := fun f hf _ => by
rcases CompleteSpace.complete hf with ⟨x, hx⟩
exact ⟨x, mem_univ x, hx⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | complete_univ | null |
CompleteSpace.prod [UniformSpace β] [CompleteSpace α] [CompleteSpace β] :
CompleteSpace (α × β) where
complete hf :=
let ⟨x1, hx1⟩ := CompleteSpace.complete <| hf.map uniformContinuous_fst
let ⟨x2, hx2⟩ := CompleteSpace.complete <| hf.map uniformContinuous_snd
⟨(x1, x2), by rw [nhds_prod_eq, le_prod]; constructor <;> assumption⟩ | instance | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | CompleteSpace.prod | null |
CompleteSpace.fst_of_prod [UniformSpace β] [CompleteSpace (α × β)] [h : Nonempty β] :
CompleteSpace α where
complete hf :=
let ⟨y⟩ := h
let ⟨(a, b), hab⟩ := CompleteSpace.complete <| hf.prod <| cauchy_pure (a := y)
⟨a, by simpa only [map_fst_prod, nhds_prod_eq] using map_mono (m := Prod.fst) hab⟩ | lemma | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | CompleteSpace.fst_of_prod | null |
CompleteSpace.snd_of_prod [UniformSpace β] [CompleteSpace (α × β)] [h : Nonempty α] :
CompleteSpace β where
complete hf :=
let ⟨x⟩ := h
let ⟨(a, b), hab⟩ := CompleteSpace.complete <| (cauchy_pure (a := x)).prod hf
⟨b, by simpa only [map_snd_prod, nhds_prod_eq] using map_mono (m := Prod.snd) hab⟩ | lemma | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | CompleteSpace.snd_of_prod | null |
completeSpace_prod_of_nonempty [UniformSpace β] [Nonempty α] [Nonempty β] :
CompleteSpace (α × β) ↔ CompleteSpace α ∧ CompleteSpace β :=
⟨fun _ ↦ ⟨.fst_of_prod (β := β), .snd_of_prod (α := α)⟩, fun ⟨_, _⟩ ↦ .prod⟩
@[to_additive] | lemma | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | completeSpace_prod_of_nonempty | null |
CompleteSpace.mulOpposite [CompleteSpace α] : CompleteSpace αᵐᵒᵖ where
complete hf :=
MulOpposite.op_surjective.exists.mpr <|
let ⟨x, hx⟩ := CompleteSpace.complete (hf.map MulOpposite.uniformContinuous_unop)
⟨x, (map_le_iff_le_comap.mp hx).trans_eq <| MulOpposite.comap_unop_nhds _⟩ | instance | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | CompleteSpace.mulOpposite | null |
completeSpace_of_isComplete_univ (h : IsComplete (univ : Set α)) : CompleteSpace α :=
⟨fun hf => let ⟨x, _, hx⟩ := h _ hf ((@principal_univ α).symm ▸ le_top); ⟨x, hx⟩⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | completeSpace_of_isComplete_univ | If `univ` is complete, the space is a complete space |
completeSpace_iff_isComplete_univ : CompleteSpace α ↔ IsComplete (univ : Set α) :=
⟨@complete_univ α _, completeSpace_of_isComplete_univ⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | completeSpace_iff_isComplete_univ | null |
completeSpace_iff_ultrafilter :
CompleteSpace α ↔ ∀ l : Ultrafilter α, Cauchy (l : Filter α) → ∃ x : α, ↑l ≤ 𝓝 x := by
simp [completeSpace_iff_isComplete_univ, isComplete_iff_ultrafilter] | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | completeSpace_iff_ultrafilter | null |
cauchy_iff_exists_le_nhds [CompleteSpace α] {l : Filter α} [NeBot l] :
Cauchy l ↔ ∃ x, l ≤ 𝓝 x :=
⟨CompleteSpace.complete, fun ⟨_, hx⟩ => cauchy_nhds.mono hx⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchy_iff_exists_le_nhds | null |
cauchy_map_iff_exists_tendsto [CompleteSpace α] {l : Filter β} {f : β → α} [NeBot l] :
Cauchy (l.map f) ↔ ∃ x, Tendsto f l (𝓝 x) :=
cauchy_iff_exists_le_nhds | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchy_map_iff_exists_tendsto | null |
cauchySeq_tendsto_of_complete [Preorder β] [CompleteSpace α] {u : β → α}
(H : CauchySeq u) : ∃ x, Tendsto u atTop (𝓝 x) :=
CompleteSpace.complete H | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchySeq_tendsto_of_complete | A Cauchy sequence in a complete space converges |
cauchySeq_tendsto_of_isComplete [Preorder β] {K : Set α} (h₁ : IsComplete K)
{u : β → α} (h₂ : ∀ n, u n ∈ K) (h₃ : CauchySeq u) : ∃ v ∈ K, Tendsto u atTop (𝓝 v) :=
h₁ _ h₃ <| le_principal_iff.2 <| mem_map_iff_exists_image.2
⟨univ, univ_mem, by rwa [image_univ, range_subset_iff]⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchySeq_tendsto_of_isComplete | If `K` is a complete subset, then any Cauchy sequence in `K` converges to a point in `K` |
Cauchy.le_nhds_lim [CompleteSpace α] {f : Filter α} (hf : Cauchy f) :
haveI := hf.1.nonempty; f ≤ 𝓝 (lim f) :=
_root_.le_nhds_lim (CompleteSpace.complete hf) | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | Cauchy.le_nhds_lim | null |
CauchySeq.tendsto_limUnder [Preorder β] [CompleteSpace α] {u : β → α} (h : CauchySeq u) :
haveI := h.1.nonempty; Tendsto u atTop (𝓝 <| limUnder atTop u) :=
h.le_nhds_lim | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | CauchySeq.tendsto_limUnder | null |
IsClosed.isComplete [CompleteSpace α] {s : Set α} (h : IsClosed s) : IsComplete s :=
fun _ cf fs =>
let ⟨x, hx⟩ := CompleteSpace.complete cf
⟨x, isClosed_iff_clusterPt.mp h x (cf.left.mono (le_inf hx fs)), hx⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | IsClosed.isComplete | null |
eq_pure_of_cauchy {f : Filter α} (hf : Cauchy f) : ∃ x : α, f = pure x := by
rcases hf with ⟨f_ne_bot, f_le⟩
simp only [DiscreteUniformity.eq_principal_idRel, le_principal_iff, mem_prod_iff] at f_le
obtain ⟨S, ⟨hS, ⟨T, ⟨hT, H⟩⟩⟩⟩ := f_le
obtain ⟨x, rfl⟩ := eq_singleton_left_of_prod_subset_idRel (f_ne_bot.nonempty_of_mem hS)
(Filter.nonempty_of_mem hT) H
exact ⟨x, f_ne_bot.le_pure_iff.mp <| le_pure_iff.mpr hS⟩
@[deprecated (since := "2025-03-23")]
alias _root_.UniformSpace.DiscreteUnif.cauchy_le_pure := eq_pure_of_cauchy | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | eq_pure_of_cauchy | A Cauchy filter in a discrete uniform space is contained in the principal filter
of a point. |
noncomputable cauchyConst {f : Filter α} (hf : Cauchy f) : α :=
(eq_pure_of_cauchy hf).choose
@[deprecated (since := "2025-03-23")]
alias _root_.UniformSpace.DiscreteUnif.cauchyConst := cauchyConst | def | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | cauchyConst | The discrete uniformity makes a space complete. -/
instance : CompleteSpace α where
complete {f} hf := by
obtain ⟨x, rfl⟩ := eq_pure_of_cauchy hf
exact ⟨x, pure_le_nhds x⟩
variable {X}
/-- A constant to which a Cauchy filter in a discrete uniform space converges. |
eq_pure_cauchyConst {f : Filter α} (hf : Cauchy f) : f = pure (cauchyConst hf) :=
(eq_pure_of_cauchy hf).choose_spec
@[deprecated (since := "2025-03-23")]
alias _root_.UniformSpace.DiscreteUnif.eq_const_of_cauchy := eq_pure_cauchyConst | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | eq_pure_cauchyConst | null |
TotallyBounded (s : Set α) : Prop :=
∀ d ∈ 𝓤 α, ∃ t : Set α, t.Finite ∧ s ⊆ ⋃ y ∈ t, { x | (x, y) ∈ d } | def | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | TotallyBounded | A set `s` is totally bounded if for every entourage `d` there is a finite
set of points `t` such that every element of `s` is `d`-near to some element of `t`. |
TotallyBounded.exists_subset {s : Set α} (hs : TotallyBounded s) {U : Set (α × α)}
(hU : U ∈ 𝓤 α) : ∃ t, t ⊆ s ∧ Set.Finite t ∧ s ⊆ ⋃ y ∈ t, { x | (x, y) ∈ U } := by
rcases comp_symm_of_uniformity hU with ⟨r, hr, rs, rU⟩
rcases hs r hr with ⟨k, fk, ks⟩
let u := k ∩ { y | ∃ x ∈ s, (x, y) ∈ r }
choose f hfs hfr using fun x : u => x.coe_prop.2
refine ⟨range f, ?_, ?_, ?_⟩
· exact range_subset_iff.2 hfs
· haveI : Fintype u := (fk.inter_of_left _).fintype
exact finite_range f
· intro x xs
obtain ⟨y, hy, xy⟩ := mem_iUnion₂.1 (ks xs)
rw [biUnion_range, mem_iUnion]
set z : ↥u := ⟨y, hy, ⟨x, xs, xy⟩⟩
exact ⟨z, rU <| mem_compRel.2 ⟨y, xy, rs (hfr z)⟩⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | TotallyBounded.exists_subset | null |
totallyBounded_iff_subset {s : Set α} :
TotallyBounded s ↔
∀ d ∈ 𝓤 α, ∃ t, t ⊆ s ∧ Set.Finite t ∧ s ⊆ ⋃ y ∈ t, { x | (x, y) ∈ d } :=
⟨fun H _ hd ↦ H.exists_subset hd, fun H d hd ↦ let ⟨t, _, ht⟩ := H d hd; ⟨t, ht⟩⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | totallyBounded_iff_subset | null |
Filter.HasBasis.totallyBounded_iff {ι} {p : ι → Prop} {U : ι → Set (α × α)}
(H : (𝓤 α).HasBasis p U) {s : Set α} :
TotallyBounded s ↔ ∀ i, p i → ∃ t : Set α, Set.Finite t ∧ s ⊆ ⋃ y ∈ t, { x | (x, y) ∈ U i } :=
H.forall_iff fun _ _ hUV h =>
h.imp fun _ ht => ⟨ht.1, ht.2.trans <| iUnion₂_mono fun _ _ _ hy => hUV hy⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | Filter.HasBasis.totallyBounded_iff | null |
totallyBounded_of_forall_symm {s : Set α}
(h : ∀ V ∈ 𝓤 α, IsSymmetricRel V → ∃ t : Set α, Set.Finite t ∧ s ⊆ ⋃ y ∈ t, ball y V) :
TotallyBounded s :=
UniformSpace.hasBasis_symmetric.totallyBounded_iff.2 fun V hV => by
simpa only [ball_eq_of_symmetry hV.2] using h V hV.1 hV.2 | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | totallyBounded_of_forall_symm | null |
TotallyBounded.subset {s₁ s₂ : Set α} (hs : s₁ ⊆ s₂) (h : TotallyBounded s₂) :
TotallyBounded s₁ := fun d hd =>
let ⟨t, ht₁, ht₂⟩ := h d hd
⟨t, ht₁, Subset.trans hs ht₂⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | TotallyBounded.subset | null |
TotallyBounded.closure {s : Set α} (h : TotallyBounded s) : TotallyBounded (closure s) :=
uniformity_hasBasis_closed.totallyBounded_iff.2 fun V hV =>
let ⟨t, htf, hst⟩ := h V hV.1
⟨t, htf, closure_minimal hst <| htf.isClosed_biUnion fun _ _ => hV.2.preimage (.prodMk_left _)⟩
@[simp] | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | TotallyBounded.closure | The closure of a totally bounded set is totally bounded. |
totallyBounded_closure {s : Set α} : TotallyBounded (closure s) ↔ TotallyBounded s :=
⟨fun h ↦ h.subset subset_closure, TotallyBounded.closure⟩ | lemma | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | totallyBounded_closure | null |
@[simp]
totallyBounded_iUnion {ι : Sort*} [Finite ι] {s : ι → Set α} :
TotallyBounded (⋃ i, s i) ↔ ∀ i, TotallyBounded (s i) := by
refine ⟨fun h i ↦ h.subset (subset_iUnion _ _), fun h U hU ↦ ?_⟩
choose t htf ht using (h · U hU)
refine ⟨⋃ i, t i, finite_iUnion htf, ?_⟩
rw [biUnion_iUnion]
gcongr; apply ht | lemma | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | totallyBounded_iUnion | A finite indexed union is totally bounded
if and only if each set of the family is totally bounded. |
totallyBounded_biUnion {ι : Type*} {I : Set ι} (hI : I.Finite) {s : ι → Set α} :
TotallyBounded (⋃ i ∈ I, s i) ↔ ∀ i ∈ I, TotallyBounded (s i) := by
have := hI.to_subtype
rw [biUnion_eq_iUnion, totallyBounded_iUnion, Subtype.forall] | lemma | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | totallyBounded_biUnion | A union indexed by a finite set is totally bounded
if and only if each set of the family is totally bounded. |
totallyBounded_sUnion {S : Set (Set α)} (hS : S.Finite) :
TotallyBounded (⋃₀ S) ↔ ∀ s ∈ S, TotallyBounded s := by
rw [sUnion_eq_biUnion, totallyBounded_biUnion hS] | lemma | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | totallyBounded_sUnion | A union of a finite family of sets is totally bounded
if and only if each set of the family is totally bounded. |
Set.Finite.totallyBounded {s : Set α} (hs : s.Finite) : TotallyBounded s := fun _U hU ↦
⟨s, hs, fun _x hx ↦ mem_biUnion hx <| refl_mem_uniformity hU⟩ | lemma | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | Set.Finite.totallyBounded | A finite set is totally bounded. |
Set.Subsingleton.totallyBounded {s : Set α} (hs : s.Subsingleton) :
TotallyBounded s :=
hs.finite.totallyBounded
@[simp] | lemma | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | Set.Subsingleton.totallyBounded | A subsingleton is totally bounded. |
totallyBounded_singleton (a : α) : TotallyBounded {a} := (finite_singleton a).totallyBounded
@[simp] | lemma | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | totallyBounded_singleton | null |
totallyBounded_empty : TotallyBounded (∅ : Set α) := finite_empty.totallyBounded | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | totallyBounded_empty | null |
@[simp]
totallyBounded_union {s t : Set α} :
TotallyBounded (s ∪ t) ↔ TotallyBounded s ∧ TotallyBounded t := by
rw [union_eq_iUnion, totallyBounded_iUnion]
simp [and_comm] | lemma | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | totallyBounded_union | The union of two sets is totally bounded
if and only if each of the two sets is totally bounded. |
protected TotallyBounded.union {s t : Set α} (hs : TotallyBounded s) (ht : TotallyBounded t) :
TotallyBounded (s ∪ t) :=
totallyBounded_union.2 ⟨hs, ht⟩
@[simp] | lemma | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | TotallyBounded.union | The union of two totally bounded sets is totally bounded. |
totallyBounded_insert (a : α) {s : Set α} :
TotallyBounded (insert a s) ↔ TotallyBounded s := by
simp_rw [← singleton_union, totallyBounded_union, totallyBounded_singleton, true_and]
protected alias ⟨_, TotallyBounded.insert⟩ := totallyBounded_insert | lemma | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | totallyBounded_insert | null |
TotallyBounded.image [UniformSpace β] {f : α → β} {s : Set α} (hs : TotallyBounded s)
(hf : UniformContinuous f) : TotallyBounded (f '' s) := fun t ht =>
have : { p : α × α | (f p.1, f p.2) ∈ t } ∈ 𝓤 α := hf ht
let ⟨c, hfc, hct⟩ := hs _ this
⟨f '' c, hfc.image f, by
simp only [mem_image, iUnion_exists, biUnion_and', iUnion_iUnion_eq_right, image_subset_iff,
preimage_iUnion, preimage_setOf_eq]
have hct : ∀ x ∈ s, ∃ i ∈ c, (f x, f i) ∈ t := by simpa [subset_def] using hct
intro x hx
simpa using hct x hx⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | TotallyBounded.image | The image of a totally bounded set under a uniformly continuous map is totally bounded. |
Ultrafilter.cauchy_of_totallyBounded {s : Set α} (f : Ultrafilter α) (hs : TotallyBounded s)
(h : ↑f ≤ 𝓟 s) : Cauchy (f : Filter α) :=
⟨f.neBot', fun _ ht =>
let ⟨t', ht'₁, ht'_symm, ht'_t⟩ := comp_symm_of_uniformity ht
let ⟨i, hi, hs_union⟩ := hs t' ht'₁
have : (⋃ y ∈ i, { x | (x, y) ∈ t' }) ∈ f := mem_of_superset (le_principal_iff.mp h) hs_union
have : ∃ y ∈ i, { x | (x, y) ∈ t' } ∈ f := (Ultrafilter.finite_biUnion_mem_iff hi).1 this
let ⟨y, _, hif⟩ := this
have : { x | (x, y) ∈ t' } ×ˢ { x | (x, y) ∈ t' } ⊆ compRel t' t' :=
fun ⟨_, _⟩ ⟨(h₁ : (_, y) ∈ t'), (h₂ : (_, y) ∈ t')⟩ => ⟨y, h₁, ht'_symm h₂⟩
mem_of_superset (prod_mem_prod hif hif) (Subset.trans this ht'_t)⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | Ultrafilter.cauchy_of_totallyBounded | null |
totallyBounded_iff_filter {s : Set α} :
TotallyBounded s ↔ ∀ f, NeBot f → f ≤ 𝓟 s → ∃ c ≤ f, Cauchy c := by
constructor
· exact fun H f hf hfs => ⟨Ultrafilter.of f, Ultrafilter.of_le f,
(Ultrafilter.of f).cauchy_of_totallyBounded H ((Ultrafilter.of_le f).trans hfs)⟩
· intro H d hd
contrapose! H with hd_cover
set f := ⨅ t : Finset α, 𝓟 (s \ ⋃ y ∈ t, { x | (x, y) ∈ d })
have hb : HasAntitoneBasis f fun t : Finset α ↦ s \ ⋃ y ∈ t, { x | (x, y) ∈ d } :=
.iInf_principal fun _ _ ↦ diff_subset_diff_right ∘ biUnion_subset_biUnion_left
have : Filter.NeBot f := hb.1.neBot_iff.2 fun _ ↦
diff_nonempty.2 <| hd_cover _ (Finset.finite_toSet _)
have : f ≤ 𝓟 s := iInf_le_of_le ∅ (by simp)
refine ⟨f, ‹_›, ‹_›, fun c hcf hc => ?_⟩
rcases mem_prod_same_iff.1 (hc.2 hd) with ⟨m, hm, hmd⟩
rcases hc.1.nonempty_of_mem hm with ⟨y, hym⟩
have : s \ {x | (x, y) ∈ d} ∈ c := by simpa using hcf (hb.mem {y})
rcases hc.1.nonempty_of_mem (inter_mem hm this) with ⟨z, hzm, -, hyz⟩
exact hyz (hmd ⟨hzm, hym⟩) | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | totallyBounded_iff_filter | null |
totallyBounded_iff_ultrafilter {s : Set α} :
TotallyBounded s ↔ ∀ f : Ultrafilter α, ↑f ≤ 𝓟 s → Cauchy (f : Filter α) := by
refine ⟨fun hs f => f.cauchy_of_totallyBounded hs, fun H => totallyBounded_iff_filter.2 ?_⟩
intro f hf hfs
exact ⟨Ultrafilter.of f, Ultrafilter.of_le f, H _ ((Ultrafilter.of_le f).trans hfs)⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | totallyBounded_iff_ultrafilter | null |
isCompact_iff_totallyBounded_isComplete {s : Set α} :
IsCompact s ↔ TotallyBounded s ∧ IsComplete s :=
⟨fun hs =>
⟨totallyBounded_iff_ultrafilter.2 fun f hf =>
let ⟨_, _, fx⟩ := isCompact_iff_ultrafilter_le_nhds.1 hs f hf
cauchy_nhds.mono fx,
fun f fc fs =>
let ⟨a, as, fa⟩ := @hs f fc.1 fs
⟨a, as, le_nhds_of_cauchy_adhp fc fa⟩⟩,
fun ⟨ht, hc⟩ =>
isCompact_iff_ultrafilter_le_nhds.2 fun f hf =>
hc _ (totallyBounded_iff_ultrafilter.1 ht f hf) hf⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | isCompact_iff_totallyBounded_isComplete | null |
protected IsCompact.totallyBounded {s : Set α} (h : IsCompact s) : TotallyBounded s :=
(isCompact_iff_totallyBounded_isComplete.1 h).1 | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | IsCompact.totallyBounded | null |
protected IsCompact.isComplete {s : Set α} (h : IsCompact s) : IsComplete s :=
(isCompact_iff_totallyBounded_isComplete.1 h).2 | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | IsCompact.isComplete | null |
TotallyBounded.isCompact_of_isComplete {s : Set α} (ht : TotallyBounded s)
(hc : IsComplete s) : IsCompact s := isCompact_iff_totallyBounded_isComplete.mpr ⟨ht, hc⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | TotallyBounded.isCompact_of_isComplete | null |
TotallyBounded.isCompact_of_isClosed [CompleteSpace α] {s : Set α} (ht : TotallyBounded s)
(hc : IsClosed s) : IsCompact s := ht.isCompact_of_isComplete hc.isComplete
@[deprecated (since := "2025-08-30")] alias isCompact_of_totallyBounded_isClosed :=
TotallyBounded.isCompact_of_isClosed | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | TotallyBounded.isCompact_of_isClosed | null |
CauchySeq.totallyBounded_range {s : ℕ → α} (hs : CauchySeq s) :
TotallyBounded (range s) := by
intro a ha
obtain ⟨n, hn⟩ := cauchySeq_iff.1 hs a ha
refine ⟨s '' { k | k ≤ n }, (finite_le_nat _).image _, ?_⟩
rw [range_subset_iff, biUnion_image]
intro m
rw [mem_iUnion₂]
rcases le_total m n with hm | hm
exacts [⟨m, hm, refl_mem_uniformity ha⟩, ⟨n, le_refl n, hn m hm n le_rfl⟩] | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | CauchySeq.totallyBounded_range | Every Cauchy sequence over `ℕ` is totally bounded. |
interUnionBalls (xs : ℕ → α) (u : ℕ → ℕ) (V : ℕ → Set (α × α)) : Set α :=
⋂ n, ⋃ m ≤ u n, UniformSpace.ball (xs m) (Prod.swap ⁻¹' V n) | def | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | interUnionBalls | Given a family of points `xs n`, a family of entourages `V n` of the diagonal and a family of
natural numbers `u n`, the intersection over `n` of the `V n`-neighborhood of `xs 1, ..., xs (u n)`.
Designed to be relatively compact when `V n` tends to the diagonal. |
totallyBounded_interUnionBalls {p : ℕ → Prop} {U : ℕ → Set (α × α)}
(H : (uniformity α).HasBasis p U) (xs : ℕ → α) (u : ℕ → ℕ) :
TotallyBounded (interUnionBalls xs u U) := by
rw [Filter.HasBasis.totallyBounded_iff H]
intro i _
have h_subset : interUnionBalls xs u U
⊆ ⋃ m ≤ u i, UniformSpace.ball (xs m) (Prod.swap ⁻¹' U i) :=
fun x hx ↦ Set.mem_iInter.1 hx i
classical
refine ⟨Finset.image xs (Finset.range (u i + 1)), Finset.finite_toSet _, fun x hx ↦ ?_⟩
simp only [Finset.coe_image, Finset.coe_range, mem_image, mem_Iio, iUnion_exists, biUnion_and',
iUnion_iUnion_eq_right, Nat.lt_succ_iff]
exact h_subset hx | lemma | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | totallyBounded_interUnionBalls | null |
isCompact_closure_interUnionBalls {p : ℕ → Prop} {U : ℕ → Set (α × α)}
(H : (uniformity α).HasBasis p U) [CompleteSpace α] (xs : ℕ → α) (u : ℕ → ℕ) :
IsCompact (closure (interUnionBalls xs u U)) := by
rw [isCompact_iff_totallyBounded_isComplete]
refine ⟨TotallyBounded.closure ?_, isClosed_closure.isComplete⟩
exact totallyBounded_interUnionBalls H xs u
/-! | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | isCompact_closure_interUnionBalls | The construction `interUnionBalls` is used to have a relatively compact set. |
setSeqAux (n : ℕ) : { s : Set α // s ∈ f ∧ s ×ˢ s ⊆ U n } :=
Classical.indefiniteDescription _ <| (cauchy_iff.1 hf).2 (U n) (U_mem n) | def | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | setSeqAux | An auxiliary sequence of sets approximating a Cauchy filter. |
setSeq (n : ℕ) : Set α :=
⋂ m ∈ Set.Iic n, (setSeqAux hf U_mem m).val | def | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | setSeq | Given a Cauchy filter `f` and a sequence `U` of entourages, `set_seq` provides
an antitone sequence of sets `s n ∈ f` such that `s n ×ˢ s n ⊆ U`. |
setSeq_mem (n : ℕ) : setSeq hf U_mem n ∈ f :=
(biInter_mem (finite_le_nat n)).2 fun m _ => (setSeqAux hf U_mem m).2.1 | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | setSeq_mem | null |
setSeq_mono ⦃m n : ℕ⦄ (h : m ≤ n) : setSeq hf U_mem n ⊆ setSeq hf U_mem m :=
biInter_subset_biInter_left <| Iic_subset_Iic.2 h | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | setSeq_mono | null |
setSeq_sub_aux (n : ℕ) : setSeq hf U_mem n ⊆ setSeqAux hf U_mem n :=
biInter_subset_of_mem right_mem_Iic | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | setSeq_sub_aux | null |
setSeq_prod_subset {N m n} (hm : N ≤ m) (hn : N ≤ n) :
setSeq hf U_mem m ×ˢ setSeq hf U_mem n ⊆ U N := fun p hp => by
refine (setSeqAux hf U_mem N).2.2 ⟨?_, ?_⟩ <;> apply setSeq_sub_aux
· exact setSeq_mono hf U_mem hm hp.1
· exact setSeq_mono hf U_mem hn hp.2 | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | setSeq_prod_subset | null |
seq (n : ℕ) : α :=
(hf.1.nonempty_of_mem (setSeq_mem hf U_mem n)).choose | def | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | seq | A sequence of points such that `seq n ∈ setSeq n`. Here `setSeq` is an antitone
sequence of sets `setSeq n ∈ f` with diameters controlled by a given sequence
of entourages. |
seq_mem (n : ℕ) : seq hf U_mem n ∈ setSeq hf U_mem n :=
(hf.1.nonempty_of_mem (setSeq_mem hf U_mem n)).choose_spec | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | seq_mem | null |
seq_pair_mem ⦃N m n : ℕ⦄ (hm : N ≤ m) (hn : N ≤ n) :
(seq hf U_mem m, seq hf U_mem n) ∈ U N :=
setSeq_prod_subset hf U_mem hm hn ⟨seq_mem hf U_mem m, seq_mem hf U_mem n⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | seq_pair_mem | null |
seq_is_cauchySeq (U_le : ∀ s ∈ 𝓤 α, ∃ n, U n ⊆ s) : CauchySeq <| seq hf U_mem :=
cauchySeq_of_controlled U U_le <| seq_pair_mem hf U_mem | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | seq_is_cauchySeq | null |
le_nhds_of_seq_tendsto_nhds (U_le : ∀ s ∈ 𝓤 α, ∃ n, U n ⊆ s)
⦃a : α⦄ (ha : Tendsto (seq hf U_mem) atTop (𝓝 a)) : f ≤ 𝓝 a :=
le_nhds_of_cauchy_adhp_aux
(fun s hs => by
rcases U_le s hs with ⟨m, hm⟩
rcases tendsto_atTop'.1 ha _ (mem_nhds_left a (U_mem m)) with ⟨n, hn⟩
refine
⟨setSeq hf U_mem (max m n), setSeq_mem hf U_mem _, ?_, seq hf U_mem (max m n), ?_,
seq_mem hf U_mem _⟩
· have := le_max_left m n
exact Set.Subset.trans (setSeq_prod_subset hf U_mem this this) hm
· exact hm (hn _ <| le_max_right m n)) | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | le_nhds_of_seq_tendsto_nhds | If the sequence `SequentiallyComplete.seq` converges to `a`, then `f ≤ 𝓝 a`. |
complete_of_convergent_controlled_sequences (U : ℕ → Set (α × α)) (U_mem : ∀ n, U n ∈ 𝓤 α)
(HU : ∀ u : ℕ → α, (∀ N m n, N ≤ m → N ≤ n → (u m, u n) ∈ U N) → ∃ a, Tendsto u atTop (𝓝 a)) :
CompleteSpace α := by
obtain ⟨U', -, hU'⟩ := (𝓤 α).exists_antitone_seq
have Hmem : ∀ n, U n ∩ U' n ∈ 𝓤 α := fun n => inter_mem (U_mem n) (hU'.2 ⟨n, Subset.refl _⟩)
refine ⟨fun hf => (HU (seq hf Hmem) fun N m n hm hn => ?_).imp <|
le_nhds_of_seq_tendsto_nhds _ _ fun s hs => ?_⟩
· exact inter_subset_left (seq_pair_mem hf Hmem hm hn)
· rcases hU'.1 hs with ⟨N, hN⟩
exact ⟨N, Subset.trans inter_subset_right hN⟩ | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | complete_of_convergent_controlled_sequences | A uniform space is complete provided that (a) its uniformity filter has a countable basis;
(b) any sequence satisfying a "controlled" version of the Cauchy condition converges. |
complete_of_cauchySeq_tendsto (H' : ∀ u : ℕ → α, CauchySeq u → ∃ a, Tendsto u atTop (𝓝 a)) :
CompleteSpace α :=
let ⟨U', _, hU'⟩ := (𝓤 α).exists_antitone_seq
complete_of_convergent_controlled_sequences U' (fun n => hU'.2 ⟨n, Subset.refl _⟩) fun u hu =>
H' u <| cauchySeq_of_controlled U' (fun _ hs => hU'.1 hs) hu
variable (α) | theorem | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | complete_of_cauchySeq_tendsto | A sequentially complete uniform space with a countable basis of the uniformity filter is
complete. |
secondCountable_of_separable [SeparableSpace α] : SecondCountableTopology α := by
rcases exists_countable_dense α with ⟨s, hsc, hsd⟩
obtain
⟨t : ℕ → Set (α × α), hto : ∀ i : ℕ, t i ∈ (𝓤 α).sets ∧ IsOpen (t i) ∧ IsSymmetricRel (t i),
h_basis : (𝓤 α).HasAntitoneBasis t⟩ :=
(@uniformity_hasBasis_open_symmetric α _).exists_antitone_subbasis
choose ht_mem hto hts using hto
refine ⟨⟨⋃ x ∈ s, range fun k => ball x (t k), hsc.biUnion fun x _ => countable_range _, ?_⟩⟩
refine (isTopologicalBasis_of_isOpen_of_nhds ?_ ?_).eq_generateFrom
· simp only [mem_iUnion₂, mem_range]
rintro _ ⟨x, _, k, rfl⟩
exact isOpen_ball x (hto k)
· intro x V hxV hVo
simp only [mem_iUnion₂, mem_range, exists_prop]
rcases UniformSpace.mem_nhds_iff.1 (IsOpen.mem_nhds hVo hxV) with ⟨U, hU, hUV⟩
rcases comp_symm_of_uniformity hU with ⟨U', hU', _, hUU'⟩
rcases h_basis.toHasBasis.mem_iff.1 hU' with ⟨k, -, hk⟩
rcases hsd.inter_open_nonempty (ball x <| t k) (isOpen_ball x (hto k))
⟨x, UniformSpace.mem_ball_self _ (ht_mem k)⟩ with
⟨y, hxy, hys⟩
refine ⟨_, ⟨y, hys, k, rfl⟩, (hts k).subset hxy, fun z hz => ?_⟩
exact hUV (ball_subset_of_comp_subset (hk hxy) hUU' (hk hz)) | instance | Topology | [
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Topology.Bases",
"Mathlib.Algebra.Order.Group.Nat",
"Mathlib.Topology.UniformSpace.DiscreteUniformity"
] | Mathlib/Topology/UniformSpace/Cauchy.lean | secondCountable_of_separable | A separable uniform space with countably generated uniformity filter is second countable:
one obtains a countable basis by taking the balls centered at points in a dense subset,
and with rational "radii" from a countable open symmetric antitone basis of `𝓤 α`. |
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