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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continuous_preStoneCechUnit : Continuous (preStoneCechUnit : α → PreStoneCech α) :=
continuous_iff_ultrafilter.mpr fun x g gx ↦ by
have : (g.map pure).toFilter ≤ 𝓝 g := by
rw [ultrafilter_converges_iff, ← bind_pure g]
rfl
have : (map preStoneCechUnit g : Filter (PreStoneCech α)) ≤ 𝓝 (Quot.mk _ g) :=
(map_mono this).trans (continuous_quot_mk.tendsto _)
convert this
exact Quot.sound ⟨x, pure_le_nhds x, gx⟩ | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | continuous_preStoneCechUnit | null |
denseRange_preStoneCechUnit : DenseRange (preStoneCechUnit : α → PreStoneCech α) :=
Quot.mk_surjective.denseRange.comp denseRange_pure continuous_coinduced_rng | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | denseRange_preStoneCechUnit | null |
preStoneCech_hom_ext {g₁ g₂ : PreStoneCech α → β} (h₁ : Continuous g₁) (h₂ : Continuous g₂)
(h : g₁ ∘ preStoneCechUnit = g₂ ∘ preStoneCechUnit) : g₁ = g₂ := by
apply Continuous.ext_on denseRange_preStoneCechUnit h₁ h₂
rintro x ⟨x, rfl⟩
apply congr_fun h x
variable [CompactSpace β]
variable {g : α → β} (hg : Continuous g)
include hg | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | preStoneCech_hom_ext | null |
preStoneCechCompat {F G : Ultrafilter α} {x : α} (hF : ↑F ≤ 𝓝 x) (hG : ↑G ≤ 𝓝 x) :
Ultrafilter.extend g F = Ultrafilter.extend g G := by
replace hF := (map_mono hF).trans hg.continuousAt
replace hG := (map_mono hG).trans hg.continuousAt
rwa [show Ultrafilter.extend g G = g x by rwa [ultrafilter_extend_eq_iff, G.coe_map],
ultrafilter_extend_eq_iff, F.coe_map] | lemma | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | preStoneCechCompat | null |
preStoneCechExtend : PreStoneCech α → β :=
Quot.lift (Ultrafilter.extend g) fun _ _ ⟨_, hF, hG⟩ ↦ preStoneCechCompat hg hF hG
@[simp] | def | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | preStoneCechExtend | The extension of a continuous function from `α` to a compact
Hausdorff space `β` to the pre-Stone-Čech compactification of `α`. |
preStoneCechExtend_extends : preStoneCechExtend hg ∘ preStoneCechUnit = g :=
ultrafilter_extend_extends g
@[simp] | lemma | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | preStoneCechExtend_extends | null |
preStoneCechExtend_preStoneCechUnit (a : α) :
preStoneCechExtend hg (preStoneCechUnit a) = g a :=
congr_fun (preStoneCechExtend_extends hg) a | lemma | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | preStoneCechExtend_preStoneCechUnit | null |
eq_if_preStoneCechUnit_eq {a b : α} (h : preStoneCechUnit a = preStoneCechUnit b) :
g a = g b := by
have e := ultrafilter_extend_extends g
rw [← congrFun e a, ← congrFun e b, Function.comp_apply, Function.comp_apply]
rw [preStoneCechUnit, preStoneCechUnit, Quot.eq] at h
generalize (pure a : Ultrafilter α) = F at h
generalize (pure b : Ultrafilter α) = G at h
induction h with
| rel x y a => exact let ⟨a, hx, hy⟩ := a; preStoneCechCompat hg hx hy
| refl x => rfl
| symm x y _ h => rw [h]
| trans x y z _ _ h h' => exact h.trans h' | lemma | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | eq_if_preStoneCechUnit_eq | null |
continuous_preStoneCechExtend : Continuous (preStoneCechExtend hg) :=
continuous_quot_lift _ (continuous_ultrafilter_extend g) | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | continuous_preStoneCechExtend | null |
StoneCech : Type u :=
T2Quotient (PreStoneCech α)
variable {α} | def | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | StoneCech | The Stone-Čech compactification of a topological space. |
stoneCechUnit (x : α) : StoneCech α :=
T2Quotient.mk (preStoneCechUnit x) | def | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | stoneCechUnit | The natural map from α to its Stone-Čech compactification. |
continuous_stoneCechUnit : Continuous (stoneCechUnit : α → StoneCech α) :=
(T2Quotient.continuous_mk _).comp continuous_preStoneCechUnit | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | continuous_stoneCechUnit | null |
denseRange_stoneCechUnit : DenseRange (stoneCechUnit : α → StoneCech α) := by
unfold stoneCechUnit T2Quotient.mk
have : Function.Surjective (T2Quotient.mk : PreStoneCech α → StoneCech α) := by
exact Quot.mk_surjective
exact this.denseRange.comp denseRange_preStoneCechUnit continuous_coinduced_rng | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | denseRange_stoneCechUnit | The image of `stoneCechUnit` is dense. (But `stoneCechUnit` need
not be an embedding, for example if the original space is not Hausdorff.) |
stoneCech_hom_ext {g₁ g₂ : StoneCech α → β} (h₁ : Continuous g₁) (h₂ : Continuous g₂)
(h : g₁ ∘ stoneCechUnit = g₂ ∘ stoneCechUnit) : g₁ = g₂ := by
apply h₁.ext_on denseRange_stoneCechUnit h₂
rintro _ ⟨x, rfl⟩
exact congr_fun h x
variable [CompactSpace β] | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | stoneCech_hom_ext | null |
stoneCechExtend : StoneCech α → β :=
T2Quotient.lift (continuous_preStoneCechExtend hg)
@[simp] | def | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | stoneCechExtend | The extension of a continuous function from `α` to a compact
Hausdorff space `β` to the Stone-Čech compactification of `α`.
This extension implements the universal property of this compactification. |
stoneCechExtend_extends : stoneCechExtend hg ∘ stoneCechUnit = g := by
ext x
rw [stoneCechExtend, Function.comp_apply, stoneCechUnit, T2Quotient.lift_mk]
apply congrFun (preStoneCechExtend_extends hg)
@[simp] | lemma | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | stoneCechExtend_extends | null |
stoneCechExtend_stoneCechUnit (a : α) : stoneCechExtend hg (stoneCechUnit a) = g a :=
congr_fun (stoneCechExtend_extends hg) a | lemma | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | stoneCechExtend_stoneCechUnit | null |
continuous_stoneCechExtend : Continuous (stoneCechExtend hg) :=
continuous_coinduced_dom.mpr (continuous_preStoneCechExtend hg) | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | continuous_stoneCechExtend | null |
eq_if_stoneCechUnit_eq {a b : α} {f : α → β} (hcf : Continuous f)
(h : stoneCechUnit a = stoneCechUnit b) : f a = f b := by
rw [← congrFun (stoneCechExtend_extends hcf), ← congrFun (stoneCechExtend_extends hcf)]
exact congrArg (stoneCechExtend hcf) h | lemma | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Connected.TotallyDisconnected"
] | Mathlib/Topology/Compactification/StoneCech.lean | eq_if_stoneCechUnit_eq | null |
eq_finite_iUnion_of_isTopologicalBasis_of_isCompact_open (b : ι → Set X)
(hb : IsTopologicalBasis (Set.range b)) (U : Set X) (hUc : IsCompact U) (hUo : IsOpen U) :
∃ s : Set ι, s.Finite ∧ U = ⋃ i ∈ s, b i := by
obtain ⟨Y, f, e, hf⟩ := hb.open_eq_iUnion hUo
choose f' hf' using hf
have : b ∘ f' = f := funext hf'
subst this
obtain ⟨t, ht⟩ :=
hUc.elim_finite_subcover (b ∘ f') (fun i => hb.isOpen (Set.mem_range_self _)) (by rw [e])
classical
refine ⟨t.image f', Set.toFinite _, le_antisymm ?_ ?_⟩
· refine Set.Subset.trans ht ?_
simp only [Set.iUnion_subset_iff]
intro i hi
simpa using subset_iUnion₂ (s := fun i _ => b (f' i)) i hi
· apply Set.iUnion₂_subset
rintro i hi
obtain ⟨j, -, rfl⟩ := Finset.mem_image.mp hi
rw [e]
exact Set.subset_iUnion (b ∘ f') j | lemma | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/Compactness/Bases.lean | eq_finite_iUnion_of_isTopologicalBasis_of_isCompact_open | null |
eq_sUnion_finset_of_isTopologicalBasis_of_isCompact_open (b : Set (Set X))
(hb : IsTopologicalBasis b) (U : Set X) (hUc : IsCompact U) (hUo : IsOpen U) :
∃ s : Finset b, U = s.toSet.sUnion := by
have hb' : b = range (fun i ↦ i : b → Set X) := by simp
rw [hb'] at hb
choose s hs hU using eq_finite_iUnion_of_isTopologicalBasis_of_isCompact_open _ hb U hUc hUo
have : Finite s := hs
let _ : Fintype s := Fintype.ofFinite _
use s.toFinset
simp [hU] | lemma | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/Compactness/Bases.lean | eq_sUnion_finset_of_isTopologicalBasis_of_isCompact_open | null |
isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis (b : ι → Set X)
(hb : IsTopologicalBasis (Set.range b)) (hb' : ∀ i, IsCompact (b i)) (U : Set X) :
IsCompact U ∧ IsOpen U ↔ ∃ s : Set ι, s.Finite ∧ U = ⋃ i ∈ s, b i := by
constructor
· exact fun ⟨h₁, h₂⟩ ↦ eq_finite_iUnion_of_isTopologicalBasis_of_isCompact_open _ hb U h₁ h₂
· rintro ⟨s, hs, rfl⟩
constructor
· exact hs.isCompact_biUnion fun i _ => hb' i
· exact isOpen_biUnion fun i _ => hb.isOpen (Set.mem_range_self _) | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.Compactness.Compact"
] | Mathlib/Topology/Compactness/Bases.lean | isCompact_open_iff_eq_finite_iUnion_of_isTopologicalBasis | 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 |
IsCompact.exists_clusterPt (hs : IsCompact s) {f : Filter X} [NeBot f] (hf : f ≤ 𝓟 s) :
∃ x ∈ s, ClusterPt x f := hs hf | lemma | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.exists_clusterPt | null |
IsCompact.exists_mapClusterPt {ι : Type*} (hs : IsCompact s) {f : Filter ι} [NeBot f]
{u : ι → X} (hf : Filter.map u f ≤ 𝓟 s) :
∃ x ∈ s, MapClusterPt x f u := hs hf | lemma | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.exists_mapClusterPt | null |
IsCompact.exists_clusterPt_of_frequently {l : Filter X} (hs : IsCompact s)
(hl : ∃ᶠ x in l, x ∈ s) : ∃ a ∈ s, ClusterPt a l :=
let ⟨a, has, ha⟩ := @hs _ (frequently_mem_iff_neBot.mp hl) inf_le_right
⟨a, has, ha.mono inf_le_left⟩ | lemma | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.exists_clusterPt_of_frequently | null |
IsCompact.exists_mapClusterPt_of_frequently {l : Filter ι} {f : ι → X} (hs : IsCompact s)
(hf : ∃ᶠ x in l, f x ∈ s) : ∃ a ∈ s, MapClusterPt a l f :=
hs.exists_clusterPt_of_frequently hf | lemma | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.exists_mapClusterPt_of_frequently | null |
IsCompact.compl_mem_sets (hs : IsCompact s) {f : Filter X} (hf : ∀ x ∈ s, sᶜ ∈ 𝓝 x ⊓ f) :
sᶜ ∈ f := by
contrapose! hf
simp only [notMem_iff_inf_principal_compl, compl_compl, inf_assoc] at hf ⊢
exact @hs _ hf inf_le_right | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.compl_mem_sets | The complement to a compact set belongs to a filter `f` if it belongs to each filter
`𝓝 x ⊓ f`, `x ∈ s`. |
IsCompact.compl_mem_sets_of_nhdsWithin (hs : IsCompact s) {f : Filter X}
(hf : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, tᶜ ∈ f) : sᶜ ∈ f := by
refine hs.compl_mem_sets fun x hx => ?_
rcases hf x hx with ⟨t, ht, hst⟩
replace ht := mem_inf_principal.1 ht
apply mem_inf_of_inter ht hst
rintro x ⟨h₁, h₂⟩ hs
exact h₂ (h₁ hs) | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.compl_mem_sets_of_nhdsWithin | The complement to a compact set belongs to a filter `f` if each `x ∈ s` has a neighborhood `t`
within `s` such that `tᶜ` belongs to `f`. |
@[elab_as_elim]
IsCompact.induction_on (hs : IsCompact s) {p : Set X → Prop} (he : p ∅)
(hmono : ∀ ⦃s t⦄, s ⊆ t → p t → p s) (hunion : ∀ ⦃s t⦄, p s → p t → p (s ∪ t))
(hnhds : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, p t) : p s := by
let f : Filter X := comk p he (fun _t ht _s hsub ↦ hmono hsub ht) (fun _s hs _t ht ↦ hunion hs ht)
have : sᶜ ∈ f := hs.compl_mem_sets_of_nhdsWithin (by simpa [f] using hnhds)
rwa [← compl_compl s] | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.induction_on | If `p : Set X → Prop` is stable under restriction and union, and each point `x`
of a compact set `s` has a neighborhood `t` within `s` such that `p t`, then `p s` holds. |
IsCompact.inter_right (hs : IsCompact s) (ht : IsClosed t) : IsCompact (s ∩ t) := by
intro f hnf hstf
obtain ⟨x, hsx, hx⟩ : ∃ x ∈ s, ClusterPt x f :=
hs (le_trans hstf (le_principal_iff.2 inter_subset_left))
have : x ∈ t := ht.mem_of_nhdsWithin_neBot <|
hx.mono <| le_trans hstf (le_principal_iff.2 inter_subset_right)
exact ⟨x, ⟨hsx, this⟩, hx⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.inter_right | The intersection of a compact set and a closed set is a compact set. |
IsCompact.inter_left (ht : IsCompact t) (hs : IsClosed s) : IsCompact (s ∩ t) :=
inter_comm t s ▸ ht.inter_right hs | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.inter_left | The intersection of a closed set and a compact set is a compact set. |
IsCompact.diff (hs : IsCompact s) (ht : IsOpen t) : IsCompact (s \ t) :=
hs.inter_right (isClosed_compl_iff.mpr ht) | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.diff | The set difference of a compact set and an open set is a compact set. |
IsCompact.of_isClosed_subset (hs : IsCompact s) (ht : IsClosed t) (h : t ⊆ s) :
IsCompact t :=
inter_eq_self_of_subset_right h ▸ hs.inter_right ht | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.of_isClosed_subset | A closed subset of a compact set is a compact set. |
IsCompact.image_of_continuousOn {f : X → Y} (hs : IsCompact s) (hf : ContinuousOn f s) :
IsCompact (f '' s) := by
intro l lne ls
have : NeBot (l.comap f ⊓ 𝓟 s) :=
comap_inf_principal_neBot_of_image_mem lne (le_principal_iff.1 ls)
obtain ⟨x, hxs, hx⟩ : ∃ x ∈ s, ClusterPt x (l.comap f ⊓ 𝓟 s) := @hs _ this inf_le_right
haveI := hx.neBot
use f x, mem_image_of_mem f hxs
have : Tendsto f (𝓝 x ⊓ (comap f l ⊓ 𝓟 s)) (𝓝 (f x) ⊓ l) := by
convert (hf x hxs).inf (@tendsto_comap _ _ f l) using 1
rw [nhdsWithin]
ac_rfl
exact this.neBot | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.image_of_continuousOn | null |
IsCompact.image {f : X → Y} (hs : IsCompact s) (hf : Continuous f) : IsCompact (f '' s) :=
hs.image_of_continuousOn hf.continuousOn | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.image | null |
IsCompact.adherence_nhdset {f : Filter X} (hs : IsCompact s) (hf₂ : f ≤ 𝓟 s)
(ht₁ : IsOpen t) (ht₂ : ∀ x ∈ s, ClusterPt x f → x ∈ t) : t ∈ f :=
Classical.by_cases mem_of_eq_bot fun (this : f ⊓ 𝓟 tᶜ ≠ ⊥) =>
let ⟨x, hx, (hfx : ClusterPt x <| f ⊓ 𝓟 tᶜ)⟩ := @hs _ ⟨this⟩ <| inf_le_of_left_le hf₂
have : x ∈ t := ht₂ x hx hfx.of_inf_left
have : tᶜ ∩ t ∈ 𝓝[tᶜ] x := inter_mem_nhdsWithin _ (IsOpen.mem_nhds ht₁ this)
have A : 𝓝[tᶜ] x = ⊥ := empty_mem_iff_bot.1 <| compl_inter_self t ▸ this
have : 𝓝[tᶜ] x ≠ ⊥ := hfx.of_inf_right.ne
absurd A this | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.adherence_nhdset | null |
isCompact_iff_ultrafilter_le_nhds :
IsCompact s ↔ ∀ f : Ultrafilter X, ↑f ≤ 𝓟 s → ∃ x ∈ s, ↑f ≤ 𝓝 x := by
refine (forall_neBot_le_iff ?_).trans ?_
· rintro f g hle ⟨x, hxs, hxf⟩
exact ⟨x, hxs, hxf.mono hle⟩
· simp only [Ultrafilter.clusterPt_iff]
alias ⟨IsCompact.ultrafilter_le_nhds, _⟩ := isCompact_iff_ultrafilter_le_nhds | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isCompact_iff_ultrafilter_le_nhds | null |
isCompact_iff_ultrafilter_le_nhds' :
IsCompact s ↔ ∀ f : Ultrafilter X, s ∈ f → ∃ x ∈ s, ↑f ≤ 𝓝 x := by
simp only [isCompact_iff_ultrafilter_le_nhds, le_principal_iff, Ultrafilter.mem_coe]
alias ⟨IsCompact.ultrafilter_le_nhds', _⟩ := isCompact_iff_ultrafilter_le_nhds' | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isCompact_iff_ultrafilter_le_nhds' | null |
IsCompact.le_nhds_of_unique_clusterPt (hs : IsCompact s) {l : Filter X} {y : X}
(hmem : s ∈ l) (h : ∀ x ∈ s, ClusterPt x l → x = y) : l ≤ 𝓝 y := by
refine le_iff_ultrafilter.2 fun f hf ↦ ?_
rcases hs.ultrafilter_le_nhds' f (hf hmem) with ⟨x, hxs, hx⟩
convert ← hx
exact h x hxs (.mono (.of_le_nhds hx) hf) | lemma | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.le_nhds_of_unique_clusterPt | If a compact set belongs to a filter and this filter has a unique cluster point `y` in this set,
then the filter is less than or equal to `𝓝 y`. |
IsCompact.tendsto_nhds_of_unique_mapClusterPt {Y} {l : Filter Y} {y : X} {f : Y → X}
(hs : IsCompact s) (hmem : ∀ᶠ x in l, f x ∈ s) (h : ∀ x ∈ s, MapClusterPt x l f → x = y) :
Tendsto f l (𝓝 y) :=
hs.le_nhds_of_unique_clusterPt (mem_map.2 hmem) h | lemma | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.tendsto_nhds_of_unique_mapClusterPt | If values of `f : Y → X` belong to a compact set `s` eventually along a filter `l`
and `y` is a unique `MapClusterPt` for `f` along `l` in `s`,
then `f` tends to `𝓝 y` along `l`. |
IsCompact.elim_directed_cover {ι : Type v} [hι : Nonempty ι] (hs : IsCompact s)
(U : ι → Set X) (hUo : ∀ i, IsOpen (U i)) (hsU : s ⊆ ⋃ i, U i) (hdU : Directed (· ⊆ ·) U) :
∃ i, s ⊆ U i :=
hι.elim fun i₀ =>
IsCompact.induction_on hs ⟨i₀, empty_subset _⟩ (fun _ _ hs ⟨i, hi⟩ => ⟨i, hs.trans hi⟩)
(fun _ _ ⟨i, hi⟩ ⟨j, hj⟩ =>
let ⟨k, hki, hkj⟩ := hdU i j
⟨k, union_subset (Subset.trans hi hki) (Subset.trans hj hkj)⟩)
fun _x hx =>
let ⟨i, hi⟩ := mem_iUnion.1 (hsU hx)
⟨U i, mem_nhdsWithin_of_mem_nhds (IsOpen.mem_nhds (hUo i) hi), i, Subset.refl _⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.elim_directed_cover | For every open directed cover of a compact set, there exists a single element of the
cover which itself includes the set. |
IsCompact.elim_finite_subcover {ι : Type v} (hs : IsCompact s) (U : ι → Set X)
(hUo : ∀ i, IsOpen (U i)) (hsU : s ⊆ ⋃ i, U i) : ∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i :=
hs.elim_directed_cover _ (fun _ => isOpen_biUnion fun i _ => hUo i)
(iUnion_eq_iUnion_finset U ▸ hsU)
(directed_of_isDirected_le fun _ _ h => biUnion_subset_biUnion_left h) | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.elim_finite_subcover | For every open cover of a compact set, there exists a finite subcover. |
IsCompact.elim_nhds_subcover_nhdsSet' (hs : IsCompact s) (U : ∀ x ∈ s, Set X)
(hU : ∀ x hx, U x hx ∈ 𝓝 x) : ∃ t : Finset s, (⋃ x ∈ t, U x.1 x.2) ∈ 𝓝ˢ s := by
rcases hs.elim_finite_subcover (fun x : s ↦ interior (U x x.2)) (fun _ ↦ isOpen_interior)
fun x hx ↦ mem_iUnion.2 ⟨⟨x, hx⟩, mem_interior_iff_mem_nhds.2 <| hU _ _⟩ with ⟨t, hst⟩
refine ⟨t, mem_nhdsSet_iff_forall.2 fun x hx ↦ ?_⟩
rcases mem_iUnion₂.1 (hst hx) with ⟨y, hyt, hy⟩
refine mem_of_superset ?_ (subset_biUnion_of_mem hyt)
exact mem_interior_iff_mem_nhds.1 hy | lemma | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.elim_nhds_subcover_nhdsSet' | null |
IsCompact.elim_nhds_subcover_nhdsSet (hs : IsCompact s) {U : X → Set X}
(hU : ∀ x ∈ s, U x ∈ 𝓝 x) : ∃ t : Finset X, (∀ x ∈ t, x ∈ s) ∧ (⋃ x ∈ t, U x) ∈ 𝓝ˢ s := by
let ⟨t, ht⟩ := hs.elim_nhds_subcover_nhdsSet' (fun x _ => U x) hU
classical
exact ⟨t.image (↑), fun x hx =>
let ⟨y, _, hyx⟩ := Finset.mem_image.1 hx
hyx ▸ y.2,
by rwa [Finset.set_biUnion_finset_image]⟩ | lemma | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.elim_nhds_subcover_nhdsSet | null |
IsCompact.elim_nhds_subcover' (hs : IsCompact s) (U : ∀ x ∈ s, Set X)
(hU : ∀ x (hx : x ∈ s), U x ‹x ∈ s› ∈ 𝓝 x) : ∃ t : Finset s, s ⊆ ⋃ x ∈ t, U (x : s) x.2 :=
(hs.elim_nhds_subcover_nhdsSet' U hU).imp fun _ ↦ subset_of_mem_nhdsSet | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.elim_nhds_subcover' | null |
IsCompact.elim_nhds_subcover (hs : IsCompact s) (U : X → Set X) (hU : ∀ x ∈ s, U x ∈ 𝓝 x) :
∃ t : Finset X, (∀ x ∈ t, x ∈ s) ∧ s ⊆ ⋃ x ∈ t, U x :=
(hs.elim_nhds_subcover_nhdsSet hU).imp fun _ h ↦ h.imp_right subset_of_mem_nhdsSet | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.elim_nhds_subcover | null |
IsCompact.elim_nhdsWithin_subcover' (hs : IsCompact s) (U : ∀ x ∈ s, Set X)
(hU : ∀ x (hx : x ∈ s), U x hx ∈ 𝓝[s] x) : ∃ t : Finset s, s ⊆ ⋃ x ∈ t, U x x.2 := by
choose V V_nhds hV using fun x hx => mem_nhdsWithin_iff_exists_mem_nhds_inter.1 (hU x hx)
refine (hs.elim_nhds_subcover' V V_nhds).imp fun t ht =>
subset_trans ?_ (iUnion₂_mono fun x _ => hV x x.2)
simpa [← iUnion_inter, ← iUnion_coe_set] | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.elim_nhdsWithin_subcover' | null |
IsCompact.elim_nhdsWithin_subcover (hs : IsCompact s) (U : X → Set X)
(hU : ∀ x ∈ s, U x ∈ 𝓝[s] x) : ∃ t : Finset X, (∀ x ∈ t, x ∈ s) ∧ s ⊆ ⋃ x ∈ t, U x := by
choose! V V_nhds hV using fun x hx => mem_nhdsWithin_iff_exists_mem_nhds_inter.1 (hU x hx)
refine (hs.elim_nhds_subcover V V_nhds).imp fun t ⟨t_sub_s, ht⟩ =>
⟨t_sub_s, subset_trans ?_ (iUnion₂_mono fun x hx => hV x (t_sub_s x hx))⟩
simpa [← iUnion_inter] | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.elim_nhdsWithin_subcover | null |
IsCompact.disjoint_nhdsSet_left {l : Filter X} (hs : IsCompact s) :
Disjoint (𝓝ˢ s) l ↔ ∀ x ∈ s, Disjoint (𝓝 x) l := by
refine ⟨fun h x hx => h.mono_left <| nhds_le_nhdsSet hx, fun H => ?_⟩
choose! U hxU hUl using fun x hx => (nhds_basis_opens x).disjoint_iff_left.1 (H x hx)
choose hxU hUo using hxU
rcases hs.elim_nhds_subcover U fun x hx => (hUo x hx).mem_nhds (hxU x hx) with ⟨t, hts, hst⟩
refine (hasBasis_nhdsSet _).disjoint_iff_left.2
⟨⋃ x ∈ t, U x, ⟨isOpen_biUnion fun x hx => hUo x (hts x hx), hst⟩, ?_⟩
rw [compl_iUnion₂, biInter_finset_mem]
exact fun x hx => hUl x (hts x hx) | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.disjoint_nhdsSet_left | The neighborhood filter of a compact set is disjoint with a filter `l` if and only if the
neighborhood filter of each point of this set is disjoint with `l`. |
IsCompact.disjoint_nhdsSet_right {l : Filter X} (hs : IsCompact s) :
Disjoint l (𝓝ˢ s) ↔ ∀ x ∈ s, Disjoint l (𝓝 x) := by
simpa only [disjoint_comm] using hs.disjoint_nhdsSet_left | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.disjoint_nhdsSet_right | A filter `l` is disjoint with the neighborhood filter of a compact set if and only if it is
disjoint with the neighborhood filter of each point of this set. |
IsCompact.elim_directed_family_closed {ι : Type v} [Nonempty ι] (hs : IsCompact s)
(t : ι → Set X) (htc : ∀ i, IsClosed (t i)) (hst : (s ∩ ⋂ i, t i) = ∅)
(hdt : Directed (· ⊇ ·) t) : ∃ i : ι, s ∩ t i = ∅ :=
let ⟨t, ht⟩ :=
hs.elim_directed_cover (compl ∘ t) (fun i => (htc i).isOpen_compl)
(by
simpa only [subset_def, not_forall, eq_empty_iff_forall_notMem, mem_iUnion, exists_prop,
mem_inter_iff, not_and, mem_iInter, mem_compl_iff] using hst)
(hdt.mono_comp _ fun _ _ => compl_subset_compl.mpr)
⟨t, by
simpa only [subset_def, not_forall, eq_empty_iff_forall_notMem, mem_iUnion, exists_prop,
mem_inter_iff, not_and, mem_iInter, mem_compl_iff] using ht⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.elim_directed_family_closed | For every directed family of closed sets whose intersection avoids a compact set,
there exists a single element of the family which itself avoids this compact set. |
IsCompact.elim_finite_subfamily_closed {ι : Type v} (hs : IsCompact s)
(t : ι → Set X) (htc : ∀ i, IsClosed (t i)) (hst : (s ∩ ⋂ i, t i) = ∅) :
∃ u : Finset ι, (s ∩ ⋂ i ∈ u, t i) = ∅ :=
hs.elim_directed_family_closed _ (fun _ ↦ isClosed_biInter fun _ _ ↦ htc _)
(by rwa [← iInter_eq_iInter_finset])
(directed_of_isDirected_le fun _ _ h ↦ biInter_subset_biInter_left h) | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.elim_finite_subfamily_closed | For every family of closed sets whose intersection avoids a compact set,
there exists a finite subfamily whose intersection avoids this compact set. |
IsCompact.inter_iInter_nonempty {ι : Type v} (hs : IsCompact s) (t : ι → Set X)
(htc : ∀ i, IsClosed (t i)) (hst : ∀ u : Finset ι, (s ∩ ⋂ i ∈ u, t i).Nonempty) :
(s ∩ ⋂ i, t i).Nonempty := by
contrapose! hst
exact hs.elim_finite_subfamily_closed t htc hst | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.inter_iInter_nonempty | To show that a compact set intersects the intersection of a family of closed sets,
it is sufficient to show that it intersects every finite subfamily. |
IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed
{ι : Type v} [hι : Nonempty ι] (t : ι → Set X) (htd : Directed (· ⊇ ·) t)
(htn : ∀ i, (t i).Nonempty) (htc : ∀ i, IsCompact (t i)) (htcl : ∀ i, IsClosed (t i)) :
(⋂ i, t i).Nonempty := by
let i₀ := hι.some
suffices (t i₀ ∩ ⋂ i, t i).Nonempty by
rwa [inter_eq_right.mpr (iInter_subset _ i₀)] at this
simp only [nonempty_iff_ne_empty] at htn ⊢
apply mt ((htc i₀).elim_directed_family_closed t htcl)
push_neg
simp only [← nonempty_iff_ne_empty] at htn ⊢
refine ⟨htd, fun i => ?_⟩
rcases htd i₀ i with ⟨j, hji₀, hji⟩
exact (htn j).mono (subset_inter hji₀ hji) | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed | Cantor's intersection theorem for `iInter`:
the intersection of a directed family of nonempty compact closed sets is nonempty. |
IsCompact.nonempty_sInter_of_directed_nonempty_isCompact_isClosed
{S : Set (Set X)} [hS : Nonempty S] (hSd : DirectedOn (· ⊇ ·) S) (hSn : ∀ U ∈ S, U.Nonempty)
(hSc : ∀ U ∈ S, IsCompact U) (hScl : ∀ U ∈ S, IsClosed U) : (⋂₀ S).Nonempty := by
rw [sInter_eq_iInter]
exact IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed _
(DirectedOn.directed_val hSd) (fun i ↦ hSn i i.2) (fun i ↦ hSc i i.2) (fun i ↦ hScl i i.2) | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.nonempty_sInter_of_directed_nonempty_isCompact_isClosed | Cantor's intersection theorem for `sInter`:
the intersection of a directed family of nonempty compact closed sets is nonempty. |
IsCompact.nonempty_iInter_of_sequence_nonempty_isCompact_isClosed (t : ℕ → Set X)
(htd : ∀ i, t (i + 1) ⊆ t i) (htn : ∀ i, (t i).Nonempty) (ht0 : IsCompact (t 0))
(htcl : ∀ i, IsClosed (t i)) : (⋂ i, t i).Nonempty :=
have tmono : Antitone t := antitone_nat_of_succ_le htd
have htd : Directed (· ⊇ ·) t := tmono.directed_ge
have : ∀ i, t i ⊆ t 0 := fun i => tmono <| Nat.zero_le i
have htc : ∀ i, IsCompact (t i) := fun i => ht0.of_isClosed_subset (htcl i) (this i)
IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed t htd htn htc htcl | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.nonempty_iInter_of_sequence_nonempty_isCompact_isClosed | Cantor's intersection theorem for sequences indexed by `ℕ`:
the intersection of a decreasing sequence of nonempty compact closed sets is nonempty. |
IsCompact.elim_finite_subcover_image {b : Set ι} {c : ι → Set X} (hs : IsCompact s)
(hc₁ : ∀ i ∈ b, IsOpen (c i)) (hc₂ : s ⊆ ⋃ i ∈ b, c i) :
∃ b', b' ⊆ b ∧ Set.Finite b' ∧ s ⊆ ⋃ i ∈ b', c i := by
simp only [Subtype.forall', biUnion_eq_iUnion] at hc₁ hc₂
rcases hs.elim_finite_subcover (fun i => c i : b → Set X) hc₁ hc₂ with ⟨d, hd⟩
refine ⟨Subtype.val '' d.toSet, ?_, d.finite_toSet.image _, ?_⟩
· simp
· rwa [biUnion_image] | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.elim_finite_subcover_image | For every open cover of a compact set, there exists a finite subcover. |
isCompact_of_finite_subcover
(h : ∀ {ι : Type u} (U : ι → Set X), (∀ i, IsOpen (U i)) → (s ⊆ ⋃ i, U i) →
∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i) :
IsCompact s := fun f hf hfs => by
contrapose! h
simp only [ClusterPt, not_neBot, ← disjoint_iff, SetCoe.forall',
(nhds_basis_opens _).disjoint_iff_left] at h
choose U hU hUf using h
refine ⟨s, U, fun x => (hU x).2, fun x hx => mem_iUnion.2 ⟨⟨x, hx⟩, (hU _).1⟩, fun t ht => ?_⟩
refine compl_notMem (le_principal_iff.1 hfs) ?_
refine mem_of_superset ((biInter_finset_mem t).2 fun x _ => hUf x) ?_
rw [subset_compl_comm, compl_iInter₂]
simpa only [compl_compl] | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isCompact_of_finite_subcover | A set `s` is compact if for every open cover of `s`, there exists a finite subcover. |
isCompact_of_finite_subfamily_closed
(h : ∀ {ι : Type u} (t : ι → Set X), (∀ i, IsClosed (t i)) → (s ∩ ⋂ i, t i) = ∅ →
∃ u : Finset ι, (s ∩ ⋂ i ∈ u, t i) = ∅) :
IsCompact s :=
isCompact_of_finite_subcover fun U hUo hsU => by
rw [← disjoint_compl_right_iff_subset, compl_iUnion, disjoint_iff] at hsU
rcases h (fun i => (U i)ᶜ) (fun i => (hUo _).isClosed_compl) hsU with ⟨t, ht⟩
refine ⟨t, ?_⟩
rwa [← disjoint_compl_right_iff_subset, compl_iUnion₂, disjoint_iff] | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isCompact_of_finite_subfamily_closed | A set `s` is compact if for every family of closed sets whose intersection avoids `s`,
there exists a finite subfamily whose intersection avoids `s`. |
isCompact_iff_finite_subcover :
IsCompact s ↔ ∀ {ι : Type u} (U : ι → Set X),
(∀ i, IsOpen (U i)) → (s ⊆ ⋃ i, U i) → ∃ t : Finset ι, s ⊆ ⋃ i ∈ t, U i :=
⟨fun hs => hs.elim_finite_subcover, isCompact_of_finite_subcover⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isCompact_iff_finite_subcover | A set `s` is compact if and only if
for every open cover of `s`, there exists a finite subcover. |
isCompact_iff_finite_subfamily_closed :
IsCompact s ↔ ∀ {ι : Type u} (t : ι → Set X),
(∀ i, IsClosed (t i)) → (s ∩ ⋂ i, t i) = ∅ → ∃ u : Finset ι, (s ∩ ⋂ i ∈ u, t i) = ∅ :=
⟨fun hs => hs.elim_finite_subfamily_closed, isCompact_of_finite_subfamily_closed⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isCompact_iff_finite_subfamily_closed | A set `s` is compact if and only if
for every family of closed sets whose intersection avoids `s`,
there exists a finite subfamily whose intersection avoids `s`. |
IsCompact.mem_nhdsSet_prod_of_forall {K : Set X} {Y} {l : Filter Y} {s : Set (X × Y)}
(hK : IsCompact K) (hs : ∀ x ∈ K, s ∈ 𝓝 x ×ˢ l) : s ∈ (𝓝ˢ K) ×ˢ l := by
refine hK.induction_on (by simp) (fun t t' ht hs ↦ ?_) (fun t t' ht ht' ↦ ?_) fun x hx ↦ ?_
· exact prod_mono (nhdsSet_mono ht) le_rfl hs
· simp [sup_prod, *]
· rcases ((nhds_basis_opens _).prod l.basis_sets).mem_iff.1 (hs x hx)
with ⟨⟨u, v⟩, ⟨⟨hx, huo⟩, hv⟩, hs⟩
refine ⟨u, nhdsWithin_le_nhds (huo.mem_nhds hx), mem_of_superset ?_ hs⟩
exact prod_mem_prod (huo.mem_nhdsSet.2 Subset.rfl) hv | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.mem_nhdsSet_prod_of_forall | If `s : Set (X × Y)` belongs to `𝓝 x ×ˢ l` for all `x` from a compact set `K`,
then it belongs to `(𝓝ˢ K) ×ˢ l`,
i.e., there exist an open `U ⊇ K` and `t ∈ l` such that `U ×ˢ t ⊆ s`. |
IsCompact.nhdsSet_prod_eq_biSup {K : Set X} (hK : IsCompact K) {Y} (l : Filter Y) :
(𝓝ˢ K) ×ˢ l = ⨆ x ∈ K, 𝓝 x ×ˢ l :=
le_antisymm (fun s hs ↦ hK.mem_nhdsSet_prod_of_forall <| by simpa using hs)
(iSup₂_le fun _ hx ↦ prod_mono (nhds_le_nhdsSet hx) le_rfl) | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.nhdsSet_prod_eq_biSup | null |
IsCompact.prod_nhdsSet_eq_biSup {K : Set Y} (hK : IsCompact K) {X} (l : Filter X) :
l ×ˢ (𝓝ˢ K) = ⨆ y ∈ K, l ×ˢ 𝓝 y := by
simp only [prod_comm (f := l), hK.nhdsSet_prod_eq_biSup, map_iSup] | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.prod_nhdsSet_eq_biSup | null |
IsCompact.mem_prod_nhdsSet_of_forall {K : Set Y} {X} {l : Filter X} {s : Set (X × Y)}
(hK : IsCompact K) (hs : ∀ y ∈ K, s ∈ l ×ˢ 𝓝 y) : s ∈ l ×ˢ 𝓝ˢ K :=
(hK.prod_nhdsSet_eq_biSup l).symm ▸ by simpa using hs | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.mem_prod_nhdsSet_of_forall | If `s : Set (X × Y)` belongs to `l ×ˢ 𝓝 y` for all `y` from a compact set `K`,
then it belongs to `l ×ˢ (𝓝ˢ K)`,
i.e., there exist `t ∈ l` and an open `U ⊇ K` such that `t ×ˢ U ⊆ s`. |
IsCompact.nhdsSet_inf_eq_biSup {K : Set X} (hK : IsCompact K) (l : Filter X) :
(𝓝ˢ K) ⊓ l = ⨆ x ∈ K, 𝓝 x ⊓ l := by
have : ∀ f : Filter X, f ⊓ l = comap (fun x ↦ (x, x)) (f ×ˢ l) := fun f ↦ by
simpa only [comap_prod] using congrArg₂ (· ⊓ ·) comap_id.symm comap_id.symm
simp_rw [this, ← comap_iSup, hK.nhdsSet_prod_eq_biSup] | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.nhdsSet_inf_eq_biSup | null |
IsCompact.inf_nhdsSet_eq_biSup {K : Set X} (hK : IsCompact K) (l : Filter X) :
l ⊓ (𝓝ˢ K) = ⨆ x ∈ K, l ⊓ 𝓝 x := by
simp only [inf_comm l, hK.nhdsSet_inf_eq_biSup] | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.inf_nhdsSet_eq_biSup | null |
IsCompact.mem_nhdsSet_inf_of_forall {K : Set X} {l : Filter X} {s : Set X}
(hK : IsCompact K) (hs : ∀ x ∈ K, s ∈ 𝓝 x ⊓ l) : s ∈ (𝓝ˢ K) ⊓ l :=
(hK.nhdsSet_inf_eq_biSup l).symm ▸ by simpa using hs | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.mem_nhdsSet_inf_of_forall | If `s : Set X` belongs to `𝓝 x ⊓ l` for all `x` from a compact set `K`,
then it belongs to `(𝓝ˢ K) ⊓ l`,
i.e., there exist an open `U ⊇ K` and `T ∈ l` such that `U ∩ T ⊆ s`. |
IsCompact.mem_inf_nhdsSet_of_forall {K : Set X} {l : Filter X} {s : Set X}
(hK : IsCompact K) (hs : ∀ y ∈ K, s ∈ l ⊓ 𝓝 y) : s ∈ l ⊓ 𝓝ˢ K :=
(hK.inf_nhdsSet_eq_biSup l).symm ▸ by simpa using hs | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.mem_inf_nhdsSet_of_forall | If `s : Set S` belongs to `l ⊓ 𝓝 x` for all `x` from a compact set `K`,
then it belongs to `l ⊓ (𝓝ˢ K)`,
i.e., there exist `T ∈ l` and an open `U ⊇ K` such that `T ∩ U ⊆ s`. |
IsCompact.eventually_forall_of_forall_eventually {x₀ : X} {K : Set Y} (hK : IsCompact K)
{P : X → Y → Prop} (hP : ∀ y ∈ K, ∀ᶠ z : X × Y in 𝓝 (x₀, y), P z.1 z.2) :
∀ᶠ x in 𝓝 x₀, ∀ y ∈ K, P x y := by
simp only [nhds_prod_eq, ← eventually_iSup, ← hK.prod_nhdsSet_eq_biSup] at hP
exact hP.curry.mono fun _ h ↦ h.self_of_nhdsSet | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.eventually_forall_of_forall_eventually | To show that `∀ y ∈ K, P x y` holds for `x` close enough to `x₀` when `K` is compact,
it is sufficient to show that for all `y₀ ∈ K` there `P x y` holds for `(x, y)` close enough
to `(x₀, y₀)`.
Provided for backwards compatibility,
see `IsCompact.mem_prod_nhdsSet_of_forall` for a stronger statement. |
isCompact_empty : IsCompact (∅ : Set X) := fun _f hnf hsf =>
Not.elim hnf.ne <| empty_mem_iff_bot.1 <| le_principal_iff.1 hsf | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isCompact_empty | null |
isCompact_singleton {x : X} : IsCompact ({x} : Set X) := fun _ hf hfa =>
⟨x, rfl, ClusterPt.of_le_nhds'
(hfa.trans <| by simpa only [principal_singleton] using pure_le_nhds x) hf⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isCompact_singleton | null |
Set.Subsingleton.isCompact (hs : s.Subsingleton) : IsCompact s :=
Subsingleton.induction_on hs isCompact_empty fun _ => isCompact_singleton | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Set.Subsingleton.isCompact | null |
Set.Finite.isCompact_biUnion {s : Set ι} {f : ι → Set X} (hs : s.Finite)
(hf : ∀ i ∈ s, IsCompact (f i)) : IsCompact (⋃ i ∈ s, f i) :=
isCompact_iff_ultrafilter_le_nhds'.2 fun l hl => by
rw [Ultrafilter.finite_biUnion_mem_iff hs] at hl
rcases hl with ⟨i, his, hi⟩
rcases (hf i his).ultrafilter_le_nhds _ (le_principal_iff.2 hi) with ⟨x, hxi, hlx⟩
exact ⟨x, mem_iUnion₂.2 ⟨i, his, hxi⟩, hlx⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Set.Finite.isCompact_biUnion | null |
Finset.isCompact_biUnion (s : Finset ι) {f : ι → Set X} (hf : ∀ i ∈ s, IsCompact (f i)) :
IsCompact (⋃ i ∈ s, f i) :=
s.finite_toSet.isCompact_biUnion hf | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Finset.isCompact_biUnion | null |
isCompact_accumulate {K : ℕ → Set X} (hK : ∀ n, IsCompact (K n)) (n : ℕ) :
IsCompact (Accumulate K n) :=
(finite_le_nat n).isCompact_biUnion fun k _ => hK k | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isCompact_accumulate | null |
Set.Finite.isCompact_sUnion {S : Set (Set X)} (hf : S.Finite) (hc : ∀ s ∈ S, IsCompact s) :
IsCompact (⋃₀ S) := by
rw [sUnion_eq_biUnion]; exact hf.isCompact_biUnion hc | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Set.Finite.isCompact_sUnion | null |
isCompact_iUnion {ι : Sort*} {f : ι → Set X} [Finite ι] (h : ∀ i, IsCompact (f i)) :
IsCompact (⋃ i, f i) :=
(finite_range f).isCompact_sUnion <| forall_mem_range.2 h
@[simp] theorem Set.Finite.isCompact (hs : s.Finite) : IsCompact s :=
biUnion_of_singleton s ▸ hs.isCompact_biUnion fun _ _ => isCompact_singleton | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isCompact_iUnion | null |
IsCompact.finite_of_discrete [DiscreteTopology X] (hs : IsCompact s) : s.Finite := by
have : ∀ x : X, ({x} : Set X) ∈ 𝓝 x := by simp [nhds_discrete]
rcases hs.elim_nhds_subcover (fun x => {x}) fun x _ => this x with ⟨t, _, hst⟩
simp only [← t.set_biUnion_coe, biUnion_of_singleton] at hst
exact t.finite_toSet.subset hst | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.finite_of_discrete | null |
isCompact_iff_finite [DiscreteTopology X] : IsCompact s ↔ s.Finite :=
⟨fun h => h.finite_of_discrete, fun h => h.isCompact⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isCompact_iff_finite | null |
IsCompact.union (hs : IsCompact s) (ht : IsCompact t) : IsCompact (s ∪ t) := by
rw [union_eq_iUnion]; exact isCompact_iUnion fun b => by cases b <;> assumption | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.union | null |
protected IsCompact.insert (hs : IsCompact s) (a) : IsCompact (insert a s) :=
isCompact_singleton.union hs | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.insert | null |
exists_subset_nhds_of_isCompact' [Nonempty ι] {V : ι → Set X}
(hV : Directed (· ⊇ ·) V) (hV_cpct : ∀ i, IsCompact (V i)) (hV_closed : ∀ i, IsClosed (V i))
{U : Set X} (hU : ∀ x ∈ ⋂ i, V i, U ∈ 𝓝 x) : ∃ i, V i ⊆ U := by
obtain ⟨W, hsubW, W_op, hWU⟩ := exists_open_set_nhds hU
suffices ∃ i, V i ⊆ W from this.imp fun i hi => hi.trans hWU
by_contra! H
replace H : ∀ i, (V i ∩ Wᶜ).Nonempty := fun i => Set.inter_compl_nonempty_iff.mpr (H i)
have : (⋂ i, V i ∩ Wᶜ).Nonempty := by
refine
IsCompact.nonempty_iInter_of_directed_nonempty_isCompact_isClosed _ (fun i j => ?_) H
(fun i => (hV_cpct i).inter_right W_op.isClosed_compl) fun i =>
(hV_closed i).inter W_op.isClosed_compl
rcases hV i j with ⟨k, hki, hkj⟩
refine ⟨k, ⟨fun x => ?_, fun x => ?_⟩⟩ <;> simp only [and_imp, mem_inter_iff, mem_compl_iff] <;>
tauto
have : ¬⋂ i : ι, V i ⊆ W := by simpa [← iInter_inter, inter_compl_nonempty_iff]
contradiction
omit [TopologicalSpace X] in | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | exists_subset_nhds_of_isCompact' | If `V : ι → Set X` is a decreasing family of closed compact sets then any neighborhood of
`⋂ i, V i` contains some `V i`. We assume each `V i` is compact *and* closed because `X` is
not assumed to be Hausdorff. See `exists_subset_nhds_of_compact` for version assuming this. |
isCompact_generateFrom [T : TopologicalSpace X]
{S : Set (Set X)} (hTS : T = generateFrom S) {s : Set X}
(h : ∀ P ⊆ S, s ⊆ ⋃₀ P → ∃ Q ⊆ P, Q.Finite ∧ s ⊆ ⋃₀ Q) :
IsCompact s := by
rw [isCompact_iff_ultrafilter_le_nhds', hTS]
intro F hsF
by_contra hF
have hSF : ∀ x ∈ s, ∃ t, x ∈ t ∧ t ∈ S ∧ t ∉ F := by simpa [nhds_generateFrom] using hF
choose! U hxU hSU hUF using hSF
obtain ⟨Q, hQU, hQ, hsQ⟩ := h (U '' s) (by simpa [Set.subset_def])
(fun x hx ↦ Set.mem_sUnion_of_mem (hxU _ hx) (by aesop))
have : ∀ s ∈ Q, s ∉ F := fun s hsQ ↦ (hQU hsQ).choose_spec.2 ▸ hUF _ (hQU hsQ).choose_spec.1
have hQF : ⋂₀ (compl '' Q) ∈ F.sets := by simpa [Filter.biInter_mem hQ, F.compl_mem_iff_notMem]
have : ⋃₀ Q ∉ F := by
simpa [-Set.sInter_image, ← Set.compl_sUnion, hsQ, F.compl_mem_iff_notMem] using hQF
exact this (F.mem_of_superset hsF hsQ) | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | isCompact_generateFrom | **Alexander's subbasis theorem**. Suppose `X` is a topological space with a subbasis `S` and `s` is
a subset of `X`. Then `s` is compact if for any open cover of `s` with all elements taken from `S`,
there is a finite subcover. |
hasBasis_cocompact : (cocompact X).HasBasis IsCompact compl :=
hasBasis_biInf_principal'
(fun s hs t ht =>
⟨s ∪ t, hs.union ht, compl_subset_compl.2 subset_union_left,
compl_subset_compl.2 subset_union_right⟩)
⟨∅, isCompact_empty⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | hasBasis_cocompact | null |
mem_cocompact : s ∈ cocompact X ↔ ∃ t, IsCompact t ∧ tᶜ ⊆ s :=
hasBasis_cocompact.mem_iff | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | mem_cocompact | null |
mem_cocompact' : s ∈ cocompact X ↔ ∃ t, IsCompact t ∧ sᶜ ⊆ t :=
mem_cocompact.trans <| exists_congr fun _ => and_congr_right fun _ => compl_subset_comm | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | mem_cocompact' | null |
_root_.IsCompact.compl_mem_cocompact (hs : IsCompact s) : sᶜ ∈ Filter.cocompact X :=
hasBasis_cocompact.mem_of_mem hs | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | _root_.IsCompact.compl_mem_cocompact | null |
cocompact_le_cofinite : cocompact X ≤ cofinite := fun s hs =>
compl_compl s ▸ hs.isCompact.compl_mem_cocompact | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | cocompact_le_cofinite | null |
cocompact_eq_cofinite (X : Type*) [TopologicalSpace X] [DiscreteTopology X] :
cocompact X = cofinite := by
simp only [cocompact, hasBasis_cofinite.eq_biInf, isCompact_iff_finite] | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | cocompact_eq_cofinite | null |
disjoint_cocompact_left (f : Filter X) :
Disjoint (Filter.cocompact X) f ↔ ∃ K ∈ f, IsCompact K := by
simp_rw [hasBasis_cocompact.disjoint_iff_left, compl_compl]
tauto | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | disjoint_cocompact_left | A filter is disjoint from the cocompact filter if and only if it contains a compact set. |
disjoint_cocompact_right (f : Filter X) :
Disjoint f (Filter.cocompact X) ↔ ∃ K ∈ f, IsCompact K := by
simp_rw [hasBasis_cocompact.disjoint_iff_right, compl_compl]
tauto | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | disjoint_cocompact_right | A filter is disjoint from the cocompact filter if and only if it contains a compact set. |
Tendsto.isCompact_insert_range_of_cocompact {f : X → Y} {y}
(hf : Tendsto f (cocompact X) (𝓝 y)) (hfc : Continuous f) : IsCompact (insert y (range f)) := by
intro l hne hle
by_cases hy : ClusterPt y l
· exact ⟨y, Or.inl rfl, hy⟩
simp only [clusterPt_iff_nonempty, not_forall, ← not_disjoint_iff_nonempty_inter, not_not] at hy
rcases hy with ⟨s, hsy, t, htl, hd⟩
rcases mem_cocompact.1 (hf hsy) with ⟨K, hKc, hKs⟩
have : f '' K ∈ l := by
filter_upwards [htl, le_principal_iff.1 hle] with y hyt hyf
rcases hyf with (rfl | ⟨x, rfl⟩)
exacts [(hd.le_bot ⟨mem_of_mem_nhds hsy, hyt⟩).elim,
mem_image_of_mem _ (not_not.1 fun hxK => hd.le_bot ⟨hKs hxK, hyt⟩)]
rcases hKc.image hfc (le_principal_iff.2 this) with ⟨y, hy, hyl⟩
exact ⟨y, Or.inr <| image_subset_range _ _ hy, hyl⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Tendsto.isCompact_insert_range_of_cocompact | null |
Tendsto.isCompact_insert_range_of_cofinite {f : ι → X} {x} (hf : Tendsto f cofinite (𝓝 x)) :
IsCompact (insert x (range f)) := by
letI : TopologicalSpace ι := ⊥; haveI h : DiscreteTopology ι := ⟨rfl⟩
rw [← cocompact_eq_cofinite ι] at hf
exact hf.isCompact_insert_range_of_cocompact continuous_of_discreteTopology | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Tendsto.isCompact_insert_range_of_cofinite | null |
Tendsto.isCompact_insert_range {f : ℕ → X} {x} (hf : Tendsto f atTop (𝓝 x)) :
IsCompact (insert x (range f)) :=
Filter.Tendsto.isCompact_insert_range_of_cofinite <| Nat.cofinite_eq_atTop.symm ▸ hf | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | Tendsto.isCompact_insert_range | null |
hasBasis_coclosedCompact :
(Filter.coclosedCompact X).HasBasis (fun s => IsClosed s ∧ IsCompact s) compl := by
simp only [Filter.coclosedCompact, iInf_and']
refine hasBasis_biInf_principal' ?_ ⟨∅, isClosed_empty, isCompact_empty⟩
rintro s ⟨hs₁, hs₂⟩ t ⟨ht₁, ht₂⟩
exact ⟨s ∪ t, ⟨⟨hs₁.union ht₁, hs₂.union ht₂⟩, compl_subset_compl.2 subset_union_left,
compl_subset_compl.2 subset_union_right⟩⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | hasBasis_coclosedCompact | null |
mem_coclosedCompact_iff :
s ∈ coclosedCompact X ↔ IsCompact (closure sᶜ) := by
refine hasBasis_coclosedCompact.mem_iff.trans ⟨?_, fun h ↦ ?_⟩
· rintro ⟨t, ⟨htcl, htco⟩, hst⟩
exact htco.of_isClosed_subset isClosed_closure <|
closure_minimal (compl_subset_comm.2 hst) htcl
· exact ⟨closure sᶜ, ⟨isClosed_closure, h⟩, compl_subset_comm.2 subset_closure⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | mem_coclosedCompact_iff | A set belongs to `coclosedCompact` if and only if the closure of its complement is compact. |
compl_mem_coclosedCompact : sᶜ ∈ coclosedCompact X ↔ IsCompact (closure s) := by
rw [mem_coclosedCompact_iff, compl_compl] | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | compl_mem_coclosedCompact | Complement of a set belongs to `coclosedCompact` if and only if its closure is compact. |
cocompact_le_coclosedCompact : cocompact X ≤ coclosedCompact X :=
iInf_mono fun _ => le_iInf fun _ => le_rfl | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | cocompact_le_coclosedCompact | null |
IsCompact.compl_mem_coclosedCompact_of_isClosed (hs : IsCompact s) (hs' : IsClosed s) :
sᶜ ∈ Filter.coclosedCompact X :=
hasBasis_coclosedCompact.mem_of_mem ⟨hs', hs⟩ | theorem | Topology | [
"Mathlib.Order.Filter.Tendsto",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Ultrafilter",
"Mathlib.Topology.Defs.Ultrafilter"
] | Mathlib/Topology/Compactness/Compact.lean | IsCompact.compl_mem_coclosedCompact_of_isClosed | null |
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