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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exists_mem (x : X) : ∃ n, x ∈ K n :=
iUnion_eq_univ_iff.1 K.iUnion_eq x | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.LocallyFinite"
] | Mathlib/Topology/Compactness/SigmaCompact.lean | exists_mem | null |
exists_mem_nhds (x : X) : ∃ n, K n ∈ 𝓝 x := by
rcases K.exists_mem x with ⟨n, hn⟩
exact ⟨n + 1, mem_interior_iff_mem_nhds.mp <| K.subset_interior_succ n hn⟩ | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.LocallyFinite"
] | Mathlib/Topology/Compactness/SigmaCompact.lean | exists_mem_nhds | null |
exists_superset_of_isCompact {s : Set X} (hs : IsCompact s) : ∃ n, s ⊆ K n := by
suffices ∃ n, s ⊆ interior (K n) from this.imp fun _ ↦ (Subset.trans · interior_subset)
refine hs.elim_directed_cover (interior ∘ K) (fun _ ↦ isOpen_interior) ?_ ?_
· intro x _
rcases K.exists_mem x with ⟨k, hk⟩
exact mem_iUnion.2 ⟨k + 1, K.subset_interior_succ _ hk⟩
· exact Monotone.directed_le fun _ _ h ↦ interior_mono <| K.subset h
open Classical in | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.LocallyFinite"
] | Mathlib/Topology/Compactness/SigmaCompact.lean | exists_superset_of_isCompact | A compact exhaustion eventually covers any compact set. |
protected noncomputable find (x : X) : ℕ :=
Nat.find (K.exists_mem x) | def | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.LocallyFinite"
] | Mathlib/Topology/Compactness/SigmaCompact.lean | find | The minimal `n` such that `x ∈ K n`. |
mem_find (x : X) : x ∈ K (K.find x) := by
classical
exact Nat.find_spec (K.exists_mem x) | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.LocallyFinite"
] | Mathlib/Topology/Compactness/SigmaCompact.lean | mem_find | null |
mem_iff_find_le {x : X} {n : ℕ} : x ∈ K n ↔ K.find x ≤ n := by
classical
exact ⟨fun h => Nat.find_min' (K.exists_mem x) h, fun h => K.subset h <| K.mem_find x⟩ | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.LocallyFinite"
] | Mathlib/Topology/Compactness/SigmaCompact.lean | mem_iff_find_le | null |
shiftr : CompactExhaustion X where
toFun n := Nat.casesOn n ∅ K
isCompact' n := Nat.casesOn n isCompact_empty K.isCompact
subset_interior_succ' n := Nat.casesOn n (empty_subset _) K.subset_interior_succ
iUnion_eq' := iUnion_eq_univ_iff.2 fun x => ⟨K.find x + 1, K.mem_find x⟩
@[simp] | def | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.LocallyFinite"
] | Mathlib/Topology/Compactness/SigmaCompact.lean | shiftr | Prepend the empty set to a compact exhaustion `K n`. |
find_shiftr (x : X) : K.shiftr.find x = K.find x + 1 := by
classical
exact Nat.find_comp_succ _ _ (notMem_empty _) | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.LocallyFinite"
] | Mathlib/Topology/Compactness/SigmaCompact.lean | find_shiftr | null |
mem_diff_shiftr_find (x : X) : x ∈ K.shiftr (K.find x + 1) \ K.shiftr (K.find x) :=
⟨K.mem_find _,
mt K.shiftr.mem_iff_find_le.1 <| by simp only [find_shiftr, not_le, Nat.lt_succ_self]⟩ | theorem | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.LocallyFinite"
] | Mathlib/Topology/Compactness/SigmaCompact.lean | mem_diff_shiftr_find | null |
noncomputable choice (X : Type*) [TopologicalSpace X] [WeaklyLocallyCompactSpace X]
[SigmaCompactSpace X] : CompactExhaustion X := by
apply Classical.choice
let K : ℕ → { s : Set X // IsCompact s } := fun n =>
Nat.recOn n ⟨∅, isCompact_empty⟩ fun n s =>
⟨(exists_compact_superset s.2).choose ∪ compactCovering X n,
(exists_compact_superset s.2).choose_spec.1.union (isCompact_compactCovering _ _)⟩
refine ⟨⟨fun n ↦ (K n).1, fun n => (K n).2, fun n ↦ ?_, ?_⟩⟩
· exact Subset.trans (exists_compact_superset (K n).2).choose_spec.2
(interior_mono subset_union_left)
· refine univ_subset_iff.1 (iUnion_compactCovering X ▸ ?_)
exact iUnion_mono' fun n => ⟨n + 1, subset_union_right⟩ | def | Topology | [
"Mathlib.Topology.Bases",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.LocallyFinite"
] | Mathlib/Topology/Compactness/SigmaCompact.lean | choice | A choice of an
[exhaustion by compact sets](https://en.wikipedia.org/wiki/Exhaustion_by_compact_sets)
of a weakly locally compact σ-compact space. |
IsPreconnected (s : Set α) : Prop :=
∀ u v : Set α, IsOpen u → IsOpen v → s ⊆ u ∪ v → (s ∩ u).Nonempty → (s ∩ v).Nonempty →
(s ∩ (u ∩ v)).Nonempty | def | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreconnected | A preconnected set is one where there is no non-trivial open partition. |
IsConnected (s : Set α) : Prop :=
s.Nonempty ∧ IsPreconnected s | def | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsConnected | A connected set is one that is nonempty and where there is no non-trivial open partition. |
IsConnected.nonempty {s : Set α} (h : IsConnected s) : s.Nonempty :=
h.1 | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsConnected.nonempty | null |
IsConnected.isPreconnected {s : Set α} (h : IsConnected s) : IsPreconnected s :=
h.2 | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsConnected.isPreconnected | null |
IsPreirreducible.isPreconnected {s : Set α} (H : IsPreirreducible s) : IsPreconnected s :=
fun _ _ hu hv _ => H _ _ hu hv | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreirreducible.isPreconnected | null |
IsIrreducible.isConnected {s : Set α} (H : IsIrreducible s) : IsConnected s :=
⟨H.nonempty, H.isPreirreducible.isPreconnected⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsIrreducible.isConnected | null |
isPreconnected_empty : IsPreconnected (∅ : Set α) :=
isPreirreducible_empty.isPreconnected | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | isPreconnected_empty | null |
isConnected_singleton {x} : IsConnected ({x} : Set α) :=
isIrreducible_singleton.isConnected | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | isConnected_singleton | null |
isPreconnected_singleton {x} : IsPreconnected ({x} : Set α) :=
isConnected_singleton.isPreconnected | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | isPreconnected_singleton | null |
Set.Subsingleton.isPreconnected {s : Set α} (hs : s.Subsingleton) : IsPreconnected s :=
hs.induction_on isPreconnected_empty fun _ => isPreconnected_singleton | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | Set.Subsingleton.isPreconnected | null |
isPreconnected_of_forall {s : Set α} (x : α)
(H : ∀ y ∈ s, ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t) : IsPreconnected s := by
rintro u v hu hv hs ⟨z, zs, zu⟩ ⟨y, ys, yv⟩
have xs : x ∈ s := by
rcases H y ys with ⟨t, ts, xt, -, -⟩
exact ts xt
cases hs xs with
| inl xu =>
rcases H y ys with ⟨t, ts, xt, yt, ht⟩
have := ht u v hu hv (ts.trans hs) ⟨x, xt, xu⟩ ⟨y, yt, yv⟩
exact this.imp fun z hz => ⟨ts hz.1, hz.2⟩
| inr xv =>
rcases H z zs with ⟨t, ts, xt, zt, ht⟩
have := ht v u hv hu (ts.trans <| by rwa [union_comm]) ⟨x, xt, xv⟩ ⟨z, zt, zu⟩
exact this.imp fun _ h => ⟨ts h.1, h.2.2, h.2.1⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | isPreconnected_of_forall | If any point of a set is joined to a fixed point by a preconnected subset,
then the original set is preconnected as well. |
isPreconnected_of_forall_pair {s : Set α}
(H : ∀ x ∈ s, ∀ y ∈ s, ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t) :
IsPreconnected s := by
rcases eq_empty_or_nonempty s with (rfl | ⟨x, hx⟩)
exacts [isPreconnected_empty, isPreconnected_of_forall x fun y => H x hx y] | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | isPreconnected_of_forall_pair | If any two points of a set are contained in a preconnected subset,
then the original set is preconnected as well. |
isPreconnected_sUnion (x : α) (c : Set (Set α)) (H1 : ∀ s ∈ c, x ∈ s)
(H2 : ∀ s ∈ c, IsPreconnected s) : IsPreconnected (⋃₀ c) := by
apply isPreconnected_of_forall x
rintro y ⟨s, sc, ys⟩
exact ⟨s, subset_sUnion_of_mem sc, H1 s sc, ys, H2 s sc⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | isPreconnected_sUnion | A union of a family of preconnected sets with a common point is preconnected as well. |
isPreconnected_iUnion {ι : Sort*} {s : ι → Set α} (h₁ : (⋂ i, s i).Nonempty)
(h₂ : ∀ i, IsPreconnected (s i)) : IsPreconnected (⋃ i, s i) :=
Exists.elim h₁ fun f hf => isPreconnected_sUnion f _ hf (forall_mem_range.2 h₂) | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | isPreconnected_iUnion | null |
IsPreconnected.union (x : α) {s t : Set α} (H1 : x ∈ s) (H2 : x ∈ t) (H3 : IsPreconnected s)
(H4 : IsPreconnected t) : IsPreconnected (s ∪ t) :=
sUnion_pair s t ▸ isPreconnected_sUnion x {s, t} (by rintro r (rfl | rfl | h) <;> assumption)
(by rintro r (rfl | rfl | h) <;> assumption) | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreconnected.union | null |
IsPreconnected.union' {s t : Set α} (H : (s ∩ t).Nonempty) (hs : IsPreconnected s)
(ht : IsPreconnected t) : IsPreconnected (s ∪ t) := by
rcases H with ⟨x, hxs, hxt⟩
exact hs.union x hxs hxt ht | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreconnected.union' | null |
IsConnected.union {s t : Set α} (H : (s ∩ t).Nonempty) (Hs : IsConnected s)
(Ht : IsConnected t) : IsConnected (s ∪ t) := by
rcases H with ⟨x, hx⟩
refine ⟨⟨x, mem_union_left t (mem_of_mem_inter_left hx)⟩, ?_⟩
exact Hs.isPreconnected.union x (mem_of_mem_inter_left hx) (mem_of_mem_inter_right hx)
Ht.isPreconnected | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsConnected.union | null |
IsPreconnected.sUnion_directed {S : Set (Set α)} (K : DirectedOn (· ⊆ ·) S)
(H : ∀ s ∈ S, IsPreconnected s) : IsPreconnected (⋃₀ S) := by
rintro u v hu hv Huv ⟨a, ⟨s, hsS, has⟩, hau⟩ ⟨b, ⟨t, htS, hbt⟩, hbv⟩
obtain ⟨r, hrS, hsr, htr⟩ : ∃ r ∈ S, s ⊆ r ∧ t ⊆ r := K s hsS t htS
have Hnuv : (r ∩ (u ∩ v)).Nonempty :=
H _ hrS u v hu hv ((subset_sUnion_of_mem hrS).trans Huv) ⟨a, hsr has, hau⟩ ⟨b, htr hbt, hbv⟩
have Kruv : r ∩ (u ∩ v) ⊆ ⋃₀ S ∩ (u ∩ v) := inter_subset_inter_left _ (subset_sUnion_of_mem hrS)
exact Hnuv.mono Kruv | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreconnected.sUnion_directed | The directed sUnion of a set S of preconnected subsets is preconnected. |
IsPreconnected.biUnion_of_reflTransGen {ι : Type*} {t : Set ι} {s : ι → Set α}
(H : ∀ i ∈ t, IsPreconnected (s i))
(K : ∀ i, i ∈ t → ∀ j, j ∈ t → ReflTransGen (fun i j => (s i ∩ s j).Nonempty ∧ i ∈ t) i j) :
IsPreconnected (⋃ n ∈ t, s n) := by
let R := fun i j : ι => (s i ∩ s j).Nonempty ∧ i ∈ t
have P : ∀ i, i ∈ t → ∀ j, j ∈ t → ReflTransGen R i j →
∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) := fun i hi j hj h => by
induction h with
| refl =>
refine ⟨{i}, singleton_subset_iff.mpr hi, mem_singleton i, mem_singleton i, ?_⟩
rw [biUnion_singleton]
exact H i hi
| @tail j k _ hjk ih =>
obtain ⟨p, hpt, hip, hjp, hp⟩ := ih hjk.2
refine ⟨insert k p, insert_subset_iff.mpr ⟨hj, hpt⟩, mem_insert_of_mem k hip,
mem_insert k p, ?_⟩
rw [biUnion_insert]
refine (H k hj).union' (hjk.1.mono ?_) hp
rw [inter_comm]
exact inter_subset_inter_right _ (subset_biUnion_of_mem hjp)
refine isPreconnected_of_forall_pair ?_
intro x hx y hy
obtain ⟨i : ι, hi : i ∈ t, hxi : x ∈ s i⟩ := mem_iUnion₂.1 hx
obtain ⟨j : ι, hj : j ∈ t, hyj : y ∈ s j⟩ := mem_iUnion₂.1 hy
obtain ⟨p, hpt, hip, hjp, hp⟩ := P i hi j hj (K i hi j hj)
exact ⟨⋃ j ∈ p, s j, biUnion_subset_biUnion_left hpt, mem_biUnion hip hxi,
mem_biUnion hjp hyj, hp⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreconnected.biUnion_of_reflTransGen | The biUnion of a family of preconnected sets is preconnected if the graph determined by
whether two sets intersect is preconnected. |
IsConnected.biUnion_of_reflTransGen {ι : Type*} {t : Set ι} {s : ι → Set α}
(ht : t.Nonempty) (H : ∀ i ∈ t, IsConnected (s i))
(K : ∀ i, i ∈ t → ∀ j, j ∈ t → ReflTransGen (fun i j : ι => (s i ∩ s j).Nonempty ∧ i ∈ t) i j) :
IsConnected (⋃ n ∈ t, s n) :=
⟨nonempty_biUnion.2 <| ⟨ht.some, ht.some_mem, (H _ ht.some_mem).nonempty⟩,
IsPreconnected.biUnion_of_reflTransGen (fun i hi => (H i hi).isPreconnected) K⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsConnected.biUnion_of_reflTransGen | The biUnion of a family of preconnected sets is preconnected if the graph determined by
whether two sets intersect is preconnected. |
IsPreconnected.iUnion_of_reflTransGen {ι : Type*} {s : ι → Set α}
(H : ∀ i, IsPreconnected (s i))
(K : ∀ i j, ReflTransGen (fun i j : ι => (s i ∩ s j).Nonempty) i j) :
IsPreconnected (⋃ n, s n) := by
rw [← biUnion_univ]
exact IsPreconnected.biUnion_of_reflTransGen (fun i _ => H i) fun i _ j _ => by
simpa [mem_univ] using K i j | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreconnected.iUnion_of_reflTransGen | Preconnectedness of the iUnion of a family of preconnected sets
indexed by the vertices of a preconnected graph,
where two vertices are joined when the corresponding sets intersect. |
IsConnected.iUnion_of_reflTransGen {ι : Type*} [Nonempty ι] {s : ι → Set α}
(H : ∀ i, IsConnected (s i))
(K : ∀ i j, ReflTransGen (fun i j : ι => (s i ∩ s j).Nonempty) i j) : IsConnected (⋃ n, s n) :=
⟨nonempty_iUnion.2 <| Nonempty.elim ‹_› fun i : ι => ⟨i, (H _).nonempty⟩,
IsPreconnected.iUnion_of_reflTransGen (fun i => (H i).isPreconnected) K⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsConnected.iUnion_of_reflTransGen | null |
IsPreconnected.iUnion_of_chain {s : β → Set α} (H : ∀ n, IsPreconnected (s n))
(K : ∀ n, (s n ∩ s (succ n)).Nonempty) : IsPreconnected (⋃ n, s n) :=
IsPreconnected.iUnion_of_reflTransGen H fun _ _ =>
reflTransGen_of_succ _ (fun i _ => K i) fun i _ => by
rw [inter_comm]
exact K i | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreconnected.iUnion_of_chain | The iUnion of connected sets indexed by a type with an archimedean successor (like `ℕ` or `ℤ`)
such that any two neighboring sets meet is preconnected. |
IsConnected.iUnion_of_chain [Nonempty β] {s : β → Set α} (H : ∀ n, IsConnected (s n))
(K : ∀ n, (s n ∩ s (succ n)).Nonempty) : IsConnected (⋃ n, s n) :=
IsConnected.iUnion_of_reflTransGen H fun _ _ =>
reflTransGen_of_succ _ (fun i _ => K i) fun i _ => by
rw [inter_comm]
exact K i | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsConnected.iUnion_of_chain | The iUnion of connected sets indexed by a type with an archimedean successor (like `ℕ` or `ℤ`)
such that any two neighboring sets meet is connected. |
IsPreconnected.biUnion_of_chain {s : β → Set α} {t : Set β} (ht : OrdConnected t)
(H : ∀ n ∈ t, IsPreconnected (s n))
(K : ∀ n : β, n ∈ t → succ n ∈ t → (s n ∩ s (succ n)).Nonempty) :
IsPreconnected (⋃ n ∈ t, s n) := by
have h1 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → k ∈ t := fun hi hj hk =>
ht.out hi hj (Ico_subset_Icc_self hk)
have h2 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → succ k ∈ t := fun hi hj hk =>
ht.out hi hj ⟨hk.1.trans <| le_succ _, succ_le_of_lt hk.2⟩
have h3 : ∀ {i j k : β}, i ∈ t → j ∈ t → k ∈ Ico i j → (s k ∩ s (succ k)).Nonempty :=
fun hi hj hk => K _ (h1 hi hj hk) (h2 hi hj hk)
refine IsPreconnected.biUnion_of_reflTransGen H fun i hi j hj => ?_
exact reflTransGen_of_succ _ (fun k hk => ⟨h3 hi hj hk, h1 hi hj hk⟩) fun k hk =>
⟨by rw [inter_comm]; exact h3 hj hi hk, h2 hj hi hk⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreconnected.biUnion_of_chain | The iUnion of preconnected sets indexed by a subset of a type with an archimedean successor
(like `ℕ` or `ℤ`) such that any two neighboring sets meet is preconnected. |
IsConnected.biUnion_of_chain {s : β → Set α} {t : Set β} (hnt : t.Nonempty)
(ht : OrdConnected t) (H : ∀ n ∈ t, IsConnected (s n))
(K : ∀ n : β, n ∈ t → succ n ∈ t → (s n ∩ s (succ n)).Nonempty) : IsConnected (⋃ n ∈ t, s n) :=
⟨nonempty_biUnion.2 <| ⟨hnt.some, hnt.some_mem, (H _ hnt.some_mem).nonempty⟩,
IsPreconnected.biUnion_of_chain ht (fun i hi => (H i hi).isPreconnected) K⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsConnected.biUnion_of_chain | The iUnion of connected sets indexed by a subset of a type with an archimedean successor
(like `ℕ` or `ℤ`) such that any two neighboring sets meet is preconnected. |
protected IsPreconnected.subset_closure {s : Set α} {t : Set α} (H : IsPreconnected s)
(Kst : s ⊆ t) (Ktcs : t ⊆ closure s) : IsPreconnected t :=
fun u v hu hv htuv ⟨_y, hyt, hyu⟩ ⟨_z, hzt, hzv⟩ =>
let ⟨p, hpu, hps⟩ := mem_closure_iff.1 (Ktcs hyt) u hu hyu
let ⟨q, hqv, hqs⟩ := mem_closure_iff.1 (Ktcs hzt) v hv hzv
let ⟨r, hrs, hruv⟩ := H u v hu hv (Subset.trans Kst htuv) ⟨p, hps, hpu⟩ ⟨q, hqs, hqv⟩
⟨r, Kst hrs, hruv⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreconnected.subset_closure | Theorem of bark and tree: if a set is within a preconnected set and its closure, then it is
preconnected as well. See also `IsConnected.subset_closure`. |
protected IsConnected.subset_closure {s : Set α} {t : Set α} (H : IsConnected s)
(Kst : s ⊆ t) (Ktcs : t ⊆ closure s) : IsConnected t :=
⟨Nonempty.mono Kst H.left, IsPreconnected.subset_closure H.right Kst Ktcs⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsConnected.subset_closure | Theorem of bark and tree: if a set is within a connected set and its closure, then it is
connected as well. See also `IsPreconnected.subset_closure`. |
protected IsPreconnected.closure {s : Set α} (H : IsPreconnected s) :
IsPreconnected (closure s) :=
IsPreconnected.subset_closure H subset_closure Subset.rfl | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreconnected.closure | The closure of a preconnected set is preconnected as well. |
protected IsConnected.closure {s : Set α} (H : IsConnected s) : IsConnected (closure s) :=
IsConnected.subset_closure H subset_closure <| Subset.rfl | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsConnected.closure | The closure of a connected set is connected as well. |
protected IsPreconnected.image [TopologicalSpace β] {s : Set α} (H : IsPreconnected s)
(f : α → β) (hf : ContinuousOn f s) : IsPreconnected (f '' s) := by
rintro u v hu hv huv ⟨_, ⟨x, xs, rfl⟩, xu⟩ ⟨_, ⟨y, ys, rfl⟩, yv⟩
rcases continuousOn_iff'.1 hf u hu with ⟨u', hu', u'_eq⟩
rcases continuousOn_iff'.1 hf v hv with ⟨v', hv', v'_eq⟩
replace huv : s ⊆ u' ∪ v' := by
rw [image_subset_iff, preimage_union] at huv
replace huv := subset_inter huv Subset.rfl
rw [union_inter_distrib_right, u'_eq, v'_eq, ← union_inter_distrib_right] at huv
exact (subset_inter_iff.1 huv).1
obtain ⟨z, hz⟩ : (s ∩ (u' ∩ v')).Nonempty := by
refine H u' v' hu' hv' huv ⟨x, ?_⟩ ⟨y, ?_⟩ <;> rw [inter_comm]
exacts [u'_eq ▸ ⟨xu, xs⟩, v'_eq ▸ ⟨yv, ys⟩]
rw [← inter_self s, inter_assoc, inter_left_comm s u', ← inter_assoc, inter_comm s, inter_comm s,
← u'_eq, ← v'_eq] at hz
exact ⟨f z, ⟨z, hz.1.2, rfl⟩, hz.1.1, hz.2.1⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreconnected.image | The image of a preconnected set is preconnected as well. |
protected IsConnected.image [TopologicalSpace β] {s : Set α} (H : IsConnected s) (f : α → β)
(hf : ContinuousOn f s) : IsConnected (f '' s) :=
⟨image_nonempty.mpr H.nonempty, H.isPreconnected.image f hf⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsConnected.image | The image of a connected set is connected as well. |
isPreconnected_closed_iff {s : Set α} :
IsPreconnected s ↔ ∀ t t', IsClosed t → IsClosed t' →
s ⊆ t ∪ t' → (s ∩ t).Nonempty → (s ∩ t').Nonempty → (s ∩ (t ∩ t')).Nonempty :=
⟨by
rintro h t t' ht ht' htt' ⟨x, xs, xt⟩ ⟨y, ys, yt'⟩
rw [← not_disjoint_iff_nonempty_inter, ← subset_compl_iff_disjoint_right, compl_inter]
intro h'
have xt' : x ∉ t' := (h' xs).resolve_left (absurd xt)
have yt : y ∉ t := (h' ys).resolve_right (absurd yt')
have := h _ _ ht.isOpen_compl ht'.isOpen_compl h' ⟨y, ys, yt⟩ ⟨x, xs, xt'⟩
rw [← compl_union] at this
exact this.ne_empty htt'.disjoint_compl_right.inter_eq,
by
rintro h u v hu hv huv ⟨x, xs, xu⟩ ⟨y, ys, yv⟩
rw [← not_disjoint_iff_nonempty_inter, ← subset_compl_iff_disjoint_right, compl_inter]
intro h'
have xv : x ∉ v := (h' xs).elim (absurd xu) id
have yu : y ∉ u := (h' ys).elim id (absurd yv)
have := h _ _ hu.isClosed_compl hv.isClosed_compl h' ⟨y, ys, yu⟩ ⟨x, xs, xv⟩
rw [← compl_union] at this
exact this.ne_empty huv.disjoint_compl_right.inter_eq⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | isPreconnected_closed_iff | null |
Topology.IsInducing.isPreconnected_image [TopologicalSpace β] {s : Set α} {f : α → β}
(hf : IsInducing f) : IsPreconnected (f '' s) ↔ IsPreconnected s := by
refine ⟨fun h => ?_, fun h => h.image _ hf.continuous.continuousOn⟩
rintro u v hu' hv' huv ⟨x, hxs, hxu⟩ ⟨y, hys, hyv⟩
rcases hf.isOpen_iff.1 hu' with ⟨u, hu, rfl⟩
rcases hf.isOpen_iff.1 hv' with ⟨v, hv, rfl⟩
replace huv : f '' s ⊆ u ∪ v := by rwa [image_subset_iff]
rcases h u v hu hv huv ⟨f x, mem_image_of_mem _ hxs, hxu⟩ ⟨f y, mem_image_of_mem _ hys, hyv⟩ with
⟨_, ⟨z, hzs, rfl⟩, hzuv⟩
exact ⟨z, hzs, hzuv⟩
/- TODO: The following lemmas about connection of preimages hold more generally for strict maps
(the quotient and subspace topologies of the image agree) whose fibers are preconnected. -/ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | Topology.IsInducing.isPreconnected_image | null |
IsPreconnected.preimage_of_isOpenMap [TopologicalSpace β] {f : α → β} {s : Set β}
(hs : IsPreconnected s) (hinj : Function.Injective f) (hf : IsOpenMap f) (hsf : s ⊆ range f) :
IsPreconnected (f ⁻¹' s) := fun u v hu hv hsuv hsu hsv => by
replace hsf : f '' (f ⁻¹' s) = s := image_preimage_eq_of_subset hsf
obtain ⟨_, has, ⟨a, hau, rfl⟩, hav⟩ : (s ∩ (f '' u ∩ f '' v)).Nonempty := by
refine hs (f '' u) (f '' v) (hf u hu) (hf v hv) ?_ ?_ ?_
· simpa only [hsf, image_union] using image_mono (f := f) hsuv
· simpa only [image_preimage_inter] using hsu.image f
· simpa only [image_preimage_inter] using hsv.image f
· exact ⟨a, has, hau, hinj.mem_set_image.1 hav⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreconnected.preimage_of_isOpenMap | null |
IsPreconnected.preimage_of_isClosedMap [TopologicalSpace β] {s : Set β}
(hs : IsPreconnected s) {f : α → β} (hinj : Function.Injective f) (hf : IsClosedMap f)
(hsf : s ⊆ range f) : IsPreconnected (f ⁻¹' s) :=
isPreconnected_closed_iff.2 fun u v hu hv hsuv hsu hsv => by
replace hsf : f '' (f ⁻¹' s) = s := image_preimage_eq_of_subset hsf
obtain ⟨_, has, ⟨a, hau, rfl⟩, hav⟩ : (s ∩ (f '' u ∩ f '' v)).Nonempty := by
refine isPreconnected_closed_iff.1 hs (f '' u) (f '' v) (hf u hu) (hf v hv) ?_ ?_ ?_
· simpa only [hsf, image_union] using image_mono (f := f) hsuv
· simpa only [image_preimage_inter] using hsu.image f
· simpa only [image_preimage_inter] using hsv.image f
· exact ⟨a, has, hau, hinj.mem_set_image.1 hav⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreconnected.preimage_of_isClosedMap | null |
IsConnected.preimage_of_isOpenMap [TopologicalSpace β] {s : Set β} (hs : IsConnected s)
{f : α → β} (hinj : Function.Injective f) (hf : IsOpenMap f) (hsf : s ⊆ range f) :
IsConnected (f ⁻¹' s) :=
⟨hs.nonempty.preimage' hsf, hs.isPreconnected.preimage_of_isOpenMap hinj hf hsf⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsConnected.preimage_of_isOpenMap | null |
IsConnected.preimage_of_isClosedMap [TopologicalSpace β] {s : Set β} (hs : IsConnected s)
{f : α → β} (hinj : Function.Injective f) (hf : IsClosedMap f) (hsf : s ⊆ range f) :
IsConnected (f ⁻¹' s) :=
⟨hs.nonempty.preimage' hsf, hs.isPreconnected.preimage_of_isClosedMap hinj hf hsf⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsConnected.preimage_of_isClosedMap | null |
IsPreconnected.subset_or_subset (hu : IsOpen u) (hv : IsOpen v) (huv : Disjoint u v)
(hsuv : s ⊆ u ∪ v) (hs : IsPreconnected s) : s ⊆ u ∨ s ⊆ v := by
specialize hs u v hu hv hsuv
obtain hsu | hsu := (s ∩ u).eq_empty_or_nonempty
· exact Or.inr ((Set.disjoint_iff_inter_eq_empty.2 hsu).subset_right_of_subset_union hsuv)
· replace hs := mt (hs hsu)
simp_rw [Set.not_nonempty_iff_eq_empty, ← Set.disjoint_iff_inter_eq_empty,
disjoint_iff_inter_eq_empty.1 huv] at hs
exact Or.inl ((hs s.disjoint_empty).subset_left_of_subset_union hsuv) | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreconnected.subset_or_subset | null |
IsPreconnected.subset_left_of_subset_union (hu : IsOpen u) (hv : IsOpen v)
(huv : Disjoint u v) (hsuv : s ⊆ u ∪ v) (hsu : (s ∩ u).Nonempty) (hs : IsPreconnected s) :
s ⊆ u :=
Disjoint.subset_left_of_subset_union hsuv
(by
by_contra hsv
rw [not_disjoint_iff_nonempty_inter] at hsv
obtain ⟨x, _, hx⟩ := hs u v hu hv hsuv hsu hsv
exact Set.disjoint_iff.1 huv hx) | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreconnected.subset_left_of_subset_union | null |
IsPreconnected.subset_right_of_subset_union (hu : IsOpen u) (hv : IsOpen v)
(huv : Disjoint u v) (hsuv : s ⊆ u ∪ v) (hsv : (s ∩ v).Nonempty) (hs : IsPreconnected s) :
s ⊆ v :=
hs.subset_left_of_subset_union hv hu huv.symm (union_comm u v ▸ hsuv) hsv | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreconnected.subset_right_of_subset_union | null |
IsPreconnected.subset_of_closure_inter_subset (hs : IsPreconnected s) (hu : IsOpen u)
(h'u : (s ∩ u).Nonempty) (h : closure u ∩ s ⊆ u) : s ⊆ u := by
have A : s ⊆ u ∪ (closure u)ᶜ := by
intro x hx
by_cases xu : x ∈ u
· exact Or.inl xu
· right
intro h'x
exact xu (h (mem_inter h'x hx))
apply hs.subset_left_of_subset_union hu isClosed_closure.isOpen_compl _ A h'u
exact disjoint_compl_right.mono_right (compl_subset_compl.2 subset_closure) | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreconnected.subset_of_closure_inter_subset | If a preconnected set `s` intersects an open set `u`, and limit points of `u` inside `s` are
contained in `u`, then the whole set `s` is contained in `u`. |
IsPreconnected.prod [TopologicalSpace β] {s : Set α} {t : Set β} (hs : IsPreconnected s)
(ht : IsPreconnected t) : IsPreconnected (s ×ˢ t) := by
apply isPreconnected_of_forall_pair
rintro ⟨a₁, b₁⟩ ⟨ha₁, hb₁⟩ ⟨a₂, b₂⟩ ⟨ha₂, hb₂⟩
refine ⟨Prod.mk a₁ '' t ∪ flip Prod.mk b₂ '' s, ?_, .inl ⟨b₁, hb₁, rfl⟩, .inr ⟨a₂, ha₂, rfl⟩, ?_⟩
· rintro _ (⟨y, hy, rfl⟩ | ⟨x, hx, rfl⟩)
exacts [⟨ha₁, hy⟩, ⟨hx, hb₂⟩]
· exact (ht.image _ (by fun_prop)).union (a₁, b₂) ⟨b₂, hb₂, rfl⟩
⟨a₁, ha₁, rfl⟩ (hs.image _ (Continuous.prodMk_left _).continuousOn) | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreconnected.prod | null |
IsConnected.prod [TopologicalSpace β] {s : Set α} {t : Set β} (hs : IsConnected s)
(ht : IsConnected t) : IsConnected (s ×ˢ t) :=
⟨hs.1.prod ht.1, hs.2.prod ht.2⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsConnected.prod | null |
isPreconnected_univ_pi [∀ i, TopologicalSpace (X i)] {s : ∀ i, Set (X i)}
(hs : ∀ i, IsPreconnected (s i)) : IsPreconnected (pi univ s) := by
rintro u v uo vo hsuv ⟨f, hfs, hfu⟩ ⟨g, hgs, hgv⟩
classical
rcases exists_finset_piecewise_mem_of_mem_nhds (uo.mem_nhds hfu) g with ⟨I, hI⟩
induction I using Finset.induction_on with
| empty =>
refine ⟨g, hgs, ⟨?_, hgv⟩⟩
simpa using hI
| insert i I _ ihI =>
rw [Finset.piecewise_insert] at hI
have := I.piecewise_mem_set_pi hfs hgs
refine (hsuv this).elim ihI fun h => ?_
set S := update (I.piecewise f g) i '' s i
have hsub : S ⊆ pi univ s := by
refine image_subset_iff.2 fun z hz => ?_
rwa [update_preimage_univ_pi]
exact fun j _ => this j trivial
have hconn : IsPreconnected S :=
(hs i).image _ (continuous_const.update i continuous_id).continuousOn
have hSu : (S ∩ u).Nonempty := ⟨_, mem_image_of_mem _ (hfs _ trivial), hI⟩
have hSv : (S ∩ v).Nonempty := ⟨_, ⟨_, this _ trivial, update_eq_self _ _⟩, h⟩
refine (hconn u v uo vo (hsub.trans hsuv) hSu hSv).mono ?_
exact inter_subset_inter_left _ hsub
@[simp] | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | isPreconnected_univ_pi | null |
isConnected_univ_pi [∀ i, TopologicalSpace (X i)] {s : ∀ i, Set (X i)} :
IsConnected (pi univ s) ↔ ∀ i, IsConnected (s i) := by
simp only [IsConnected, ← univ_pi_nonempty_iff, forall_and, and_congr_right_iff]
refine fun hne => ⟨fun hc i => ?_, isPreconnected_univ_pi⟩
rw [← eval_image_univ_pi hne]
exact hc.image _ (continuous_apply _).continuousOn | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | isConnected_univ_pi | null |
connectedComponent (x : α) : Set α :=
⋃₀ { s : Set α | IsPreconnected s ∧ x ∈ s }
open Classical in | def | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | connectedComponent | The connected component of a point is the maximal connected set
that contains this point. |
connectedComponentIn (F : Set α) (x : α) : Set α :=
if h : x ∈ F then (↑) '' connectedComponent (⟨x, h⟩ : F) else ∅ | def | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | connectedComponentIn | Given a set `F` in a topological space `α` and a point `x : α`, the connected
component of `x` in `F` is the connected component of `x` in the subtype `F` seen as
a set in `α`. This definition does not make sense if `x` is not in `F` so we return the
empty set in this case. |
connectedComponentIn_eq_image {F : Set α} {x : α} (h : x ∈ F) :
connectedComponentIn F x = (↑) '' connectedComponent (⟨x, h⟩ : F) :=
dif_pos h | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | connectedComponentIn_eq_image | null |
connectedComponentIn_eq_empty {F : Set α} {x : α} (h : x ∉ F) :
connectedComponentIn F x = ∅ :=
dif_neg h | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | connectedComponentIn_eq_empty | null |
mem_connectedComponent {x : α} : x ∈ connectedComponent x :=
mem_sUnion_of_mem (mem_singleton x) ⟨isPreconnected_singleton, mem_singleton x⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | mem_connectedComponent | null |
mem_connectedComponentIn {x : α} {F : Set α} (hx : x ∈ F) :
x ∈ connectedComponentIn F x := by
simp [connectedComponentIn_eq_image hx, mem_connectedComponent, hx] | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | mem_connectedComponentIn | null |
connectedComponent_nonempty {x : α} : (connectedComponent x).Nonempty :=
⟨x, mem_connectedComponent⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | connectedComponent_nonempty | null |
connectedComponentIn_nonempty_iff {x : α} {F : Set α} :
(connectedComponentIn F x).Nonempty ↔ x ∈ F := by
rw [connectedComponentIn]
split_ifs <;> simp [connectedComponent_nonempty, *] | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | connectedComponentIn_nonempty_iff | null |
connectedComponentIn_subset (F : Set α) (x : α) : connectedComponentIn F x ⊆ F := by
rw [connectedComponentIn]
split_ifs <;> simp | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | connectedComponentIn_subset | null |
isPreconnected_connectedComponent {x : α} : IsPreconnected (connectedComponent x) :=
isPreconnected_sUnion x _ (fun _ => And.right) fun _ => And.left | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | isPreconnected_connectedComponent | null |
isPreconnected_connectedComponentIn {x : α} {F : Set α} :
IsPreconnected (connectedComponentIn F x) := by
rw [connectedComponentIn]; split_ifs
· exact IsInducing.subtypeVal.isPreconnected_image.mpr isPreconnected_connectedComponent
· exact isPreconnected_empty | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | isPreconnected_connectedComponentIn | null |
isConnected_connectedComponent {x : α} : IsConnected (connectedComponent x) :=
⟨⟨x, mem_connectedComponent⟩, isPreconnected_connectedComponent⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | isConnected_connectedComponent | null |
isConnected_connectedComponentIn_iff {x : α} {F : Set α} :
IsConnected (connectedComponentIn F x) ↔ x ∈ F := by
simp_rw [← connectedComponentIn_nonempty_iff, IsConnected, isPreconnected_connectedComponentIn,
and_true] | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | isConnected_connectedComponentIn_iff | null |
IsPreconnected.subset_connectedComponent {x : α} {s : Set α} (H1 : IsPreconnected s)
(H2 : x ∈ s) : s ⊆ connectedComponent x := fun _z hz => mem_sUnion_of_mem hz ⟨H1, H2⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreconnected.subset_connectedComponent | null |
IsPreconnected.subset_connectedComponentIn {x : α} {F : Set α} (hs : IsPreconnected s)
(hxs : x ∈ s) (hsF : s ⊆ F) : s ⊆ connectedComponentIn F x := by
have : IsPreconnected (((↑) : F → α) ⁻¹' s) := by
refine IsInducing.subtypeVal.isPreconnected_image.mp ?_
rwa [Subtype.image_preimage_coe, inter_eq_right.mpr hsF]
have h2xs : (⟨x, hsF hxs⟩ : F) ∈ (↑) ⁻¹' s := by
rw [mem_preimage]
exact hxs
have := this.subset_connectedComponent h2xs
rw [connectedComponentIn_eq_image (hsF hxs)]
refine Subset.trans ?_ (image_mono this)
rw [Subtype.image_preimage_coe, inter_eq_right.mpr hsF] | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreconnected.subset_connectedComponentIn | null |
IsConnected.subset_connectedComponent {x : α} {s : Set α} (H1 : IsConnected s)
(H2 : x ∈ s) : s ⊆ connectedComponent x :=
H1.2.subset_connectedComponent H2 | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsConnected.subset_connectedComponent | null |
IsPreconnected.connectedComponentIn {x : α} {F : Set α} (h : IsPreconnected F)
(hx : x ∈ F) : connectedComponentIn F x = F :=
(connectedComponentIn_subset F x).antisymm (h.subset_connectedComponentIn hx subset_rfl) | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | IsPreconnected.connectedComponentIn | null |
connectedComponent_eq {x y : α} (h : y ∈ connectedComponent x) :
connectedComponent x = connectedComponent y :=
eq_of_subset_of_subset (isConnected_connectedComponent.subset_connectedComponent h)
(isConnected_connectedComponent.subset_connectedComponent
(Set.mem_of_mem_of_subset mem_connectedComponent
(isConnected_connectedComponent.subset_connectedComponent h))) | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | connectedComponent_eq | null |
connectedComponent_eq_iff_mem {x y : α} :
connectedComponent x = connectedComponent y ↔ x ∈ connectedComponent y :=
⟨fun h => h ▸ mem_connectedComponent, fun h => (connectedComponent_eq h).symm⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | connectedComponent_eq_iff_mem | null |
connectedComponentIn_eq {x y : α} {F : Set α} (h : y ∈ connectedComponentIn F x) :
connectedComponentIn F x = connectedComponentIn F y := by
have hx : x ∈ F := connectedComponentIn_nonempty_iff.mp ⟨y, h⟩
simp_rw [connectedComponentIn_eq_image hx] at h ⊢
obtain ⟨⟨y, hy⟩, h2y, rfl⟩ := h
simp_rw [connectedComponentIn_eq_image hy, connectedComponent_eq h2y] | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | connectedComponentIn_eq | null |
connectedComponentIn_univ (x : α) : connectedComponentIn univ x = connectedComponent x :=
subset_antisymm
(isPreconnected_connectedComponentIn.subset_connectedComponent <|
mem_connectedComponentIn trivial)
(isPreconnected_connectedComponent.subset_connectedComponentIn mem_connectedComponent <|
subset_univ _) | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | connectedComponentIn_univ | null |
connectedComponent_disjoint {x y : α} (h : connectedComponent x ≠ connectedComponent y) :
Disjoint (connectedComponent x) (connectedComponent y) :=
Set.disjoint_left.2 fun _ h1 h2 =>
h ((connectedComponent_eq h1).trans (connectedComponent_eq h2).symm) | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | connectedComponent_disjoint | null |
isClosed_connectedComponent {x : α} : IsClosed (connectedComponent x) :=
closure_subset_iff_isClosed.1 <|
isConnected_connectedComponent.closure.subset_connectedComponent <|
subset_closure mem_connectedComponent | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | isClosed_connectedComponent | null |
Continuous.image_connectedComponent_subset [TopologicalSpace β] {f : α → β}
(h : Continuous f) (a : α) : f '' connectedComponent a ⊆ connectedComponent (f a) :=
(isConnected_connectedComponent.image f h.continuousOn).subset_connectedComponent
((mem_image f (connectedComponent a) (f a)).2 ⟨a, mem_connectedComponent, rfl⟩) | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | Continuous.image_connectedComponent_subset | null |
Continuous.image_connectedComponentIn_subset [TopologicalSpace β] {f : α → β} {s : Set α}
{a : α} (hf : Continuous f) (hx : a ∈ s) :
f '' connectedComponentIn s a ⊆ connectedComponentIn (f '' s) (f a) :=
(isPreconnected_connectedComponentIn.image _ hf.continuousOn).subset_connectedComponentIn
(mem_image_of_mem _ <| mem_connectedComponentIn hx)
(image_mono <| connectedComponentIn_subset _ _) | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | Continuous.image_connectedComponentIn_subset | null |
Continuous.mapsTo_connectedComponent [TopologicalSpace β] {f : α → β} (h : Continuous f)
(a : α) : MapsTo f (connectedComponent a) (connectedComponent (f a)) :=
mapsTo_iff_image_subset.2 <| h.image_connectedComponent_subset a | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | Continuous.mapsTo_connectedComponent | null |
Continuous.mapsTo_connectedComponentIn [TopologicalSpace β] {f : α → β} {s : Set α}
(h : Continuous f) {a : α} (hx : a ∈ s) :
MapsTo f (connectedComponentIn s a) (connectedComponentIn (f '' s) (f a)) :=
mapsTo_iff_image_subset.2 <| image_connectedComponentIn_subset h hx | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | Continuous.mapsTo_connectedComponentIn | null |
irreducibleComponent_subset_connectedComponent {x : α} :
irreducibleComponent x ⊆ connectedComponent x :=
isIrreducible_irreducibleComponent.isConnected.subset_connectedComponent mem_irreducibleComponent
@[mono] | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | irreducibleComponent_subset_connectedComponent | null |
connectedComponentIn_mono (x : α) {F G : Set α} (h : F ⊆ G) :
connectedComponentIn F x ⊆ connectedComponentIn G x := by
by_cases hx : x ∈ F
· rw [connectedComponentIn_eq_image hx, connectedComponentIn_eq_image (h hx), ←
show ((↑) : G → α) ∘ inclusion h = (↑) from rfl, image_comp]
exact image_mono ((continuous_inclusion h).image_connectedComponent_subset ⟨x, hx⟩)
· rw [connectedComponentIn_eq_empty hx]
exact Set.empty_subset _ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | connectedComponentIn_mono | null |
PreconnectedSpace (α : Type u) [TopologicalSpace α] : Prop where
/-- The universal set `Set.univ` in a preconnected space is a preconnected set. -/
isPreconnected_univ : IsPreconnected (univ : Set α)
export PreconnectedSpace (isPreconnected_univ) | class | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | PreconnectedSpace | A preconnected space is one where there is no non-trivial open partition. |
ConnectedSpace (α : Type u) [TopologicalSpace α] : Prop extends PreconnectedSpace α where
/-- A connected space is nonempty. -/
toNonempty : Nonempty α
attribute [instance 50] ConnectedSpace.toNonempty -- see Note [lower instance priority] | class | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | ConnectedSpace | A connected space is a nonempty one where there is no non-trivial open partition. |
isConnected_univ [ConnectedSpace α] : IsConnected (univ : Set α) :=
⟨univ_nonempty, isPreconnected_univ⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | isConnected_univ | null |
preconnectedSpace_iff_univ : PreconnectedSpace α ↔ IsPreconnected (univ : Set α) :=
⟨fun h ↦ h.1, fun h ↦ ⟨h⟩⟩ | lemma | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | preconnectedSpace_iff_univ | null |
connectedSpace_iff_univ : ConnectedSpace α ↔ IsConnected (univ : Set α) :=
⟨fun h ↦ ⟨univ_nonempty, h.1.1⟩,
fun h ↦ ConnectedSpace.mk (toPreconnectedSpace := ⟨h.2⟩) ⟨h.1.some⟩⟩ | lemma | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | connectedSpace_iff_univ | null |
isPreconnected_range [TopologicalSpace β] [PreconnectedSpace α] {f : α → β}
(h : Continuous f) : IsPreconnected (range f) :=
@image_univ _ _ f ▸ isPreconnected_univ.image _ h.continuousOn | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | isPreconnected_range | null |
isConnected_range [TopologicalSpace β] [ConnectedSpace α] {f : α → β} (h : Continuous f) :
IsConnected (range f) :=
⟨range_nonempty f, isPreconnected_range h⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | isConnected_range | null |
Function.Surjective.connectedSpace [ConnectedSpace α] [TopologicalSpace β]
{f : α → β} (hf : Surjective f) (hf' : Continuous f) : ConnectedSpace β := by
rw [connectedSpace_iff_univ, ← hf.range_eq]
exact isConnected_range hf' | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | Function.Surjective.connectedSpace | null |
Quotient.instConnectedSpace {s : Setoid α} [ConnectedSpace α] :
ConnectedSpace (Quotient s) :=
Quotient.mk'_surjective.connectedSpace continuous_coinduced_rng | instance | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | Quotient.instConnectedSpace | null |
DenseRange.preconnectedSpace [TopologicalSpace β] [PreconnectedSpace α] {f : α → β}
(hf : DenseRange f) (hc : Continuous f) : PreconnectedSpace β :=
⟨hf.closure_eq ▸ (isPreconnected_range hc).closure⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | DenseRange.preconnectedSpace | null |
connectedSpace_iff_connectedComponent :
ConnectedSpace α ↔ ∃ x : α, connectedComponent x = univ := by
constructor
· rintro ⟨⟨x⟩⟩
exact
⟨x, eq_univ_of_univ_subset <| isPreconnected_univ.subset_connectedComponent (mem_univ x)⟩
· rintro ⟨x, h⟩
haveI : PreconnectedSpace α :=
⟨by rw [← h]; exact isPreconnected_connectedComponent⟩
exact ⟨⟨x⟩⟩ | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | connectedSpace_iff_connectedComponent | null |
preconnectedSpace_iff_connectedComponent :
PreconnectedSpace α ↔ ∀ x : α, connectedComponent x = univ := by
constructor
· intro h x
exact eq_univ_of_univ_subset <| isPreconnected_univ.subset_connectedComponent (mem_univ x)
· intro h
rcases isEmpty_or_nonempty α with hα | hα
· exact ⟨by rw [univ_eq_empty_iff.mpr hα]; exact isPreconnected_empty⟩
· exact ⟨by rw [← h (Classical.choice hα)]; exact isPreconnected_connectedComponent⟩
@[simp] | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | preconnectedSpace_iff_connectedComponent | null |
PreconnectedSpace.connectedComponent_eq_univ {X : Type*} [TopologicalSpace X]
[h : PreconnectedSpace X] (x : X) : connectedComponent x = univ :=
preconnectedSpace_iff_connectedComponent.mp h x | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | PreconnectedSpace.connectedComponent_eq_univ | null |
Subtype.preconnectedSpace {s : Set α} (h : IsPreconnected s) : PreconnectedSpace s where
isPreconnected_univ := by
rwa [← IsInducing.subtypeVal.isPreconnected_image, image_univ, Subtype.range_val] | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | Subtype.preconnectedSpace | null |
Subtype.connectedSpace {s : Set α} (h : IsConnected s) : ConnectedSpace s where
toPreconnectedSpace := Subtype.preconnectedSpace h.isPreconnected
toNonempty := h.nonempty.to_subtype | theorem | Topology | [
"Mathlib.Data.Set.SymmDiff",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Topology.Irreducible"
] | Mathlib/Topology/Connected/Basic.lean | Subtype.connectedSpace | null |
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