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sigma.is_preconnected_iff [hι : nonempty ι] [Π i, topological_space (π i)]
{s : set (Σ i, π i)} :
is_preconnected s ↔ ∃ i t, is_preconnected t ∧ s = sigma.mk i '' t | begin
refine ⟨λ hs, _, _⟩,
{ obtain rfl | h := s.eq_empty_or_nonempty,
{ exact ⟨classical.choice hι, ∅, is_preconnected_empty, (set.image_empty _).symm⟩ },
{ obtain ⟨a, t, ht, rfl⟩ := sigma.is_connected_iff.1 ⟨h, hs⟩,
refine ⟨a, t, ht.is_preconnected, rfl⟩ } },
{ rintro ⟨a, t, ht, rfl⟩,
exact ht... | lemma | sigma.is_preconnected_iff | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected",
"is_preconnected_empty",
"set.image_empty",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum.is_connected_iff [topological_space β] {s : set (α ⊕ β)} :
is_connected s ↔
(∃ t, is_connected t ∧ s = sum.inl '' t) ∨ ∃ t, is_connected t ∧ s = sum.inr '' t | begin
refine ⟨λ hs, _, _⟩,
{ let u : set (α ⊕ β) := range sum.inl,
let v : set (α ⊕ β) := range sum.inr,
have hu : is_open u, exact is_open_range_inl,
obtain ⟨x | x, hx⟩ := hs.nonempty,
{ have h : s ⊆ range sum.inl := is_preconnected.subset_left_of_subset_union
is_open_range_inl is_open_rang... | lemma | sum.is_connected_iff | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_connected",
"is_open",
"is_open_range_inl",
"is_open_range_inr",
"is_preconnected.subset_left_of_subset_union",
"is_preconnected.subset_right_of_subset_union",
"set.image_preimage_eq_of_subset",
"sum.inl_injective",
"sum.inr_injective",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
sum.is_preconnected_iff [topological_space β] {s : set (α ⊕ β)} :
is_preconnected s ↔
(∃ t, is_preconnected t ∧ s = sum.inl '' t) ∨ ∃ t, is_preconnected t ∧ s = sum.inr '' t | begin
refine ⟨λ hs, _, _⟩,
{ obtain rfl | h := s.eq_empty_or_nonempty,
{ exact or.inl ⟨∅, is_preconnected_empty, (set.image_empty _).symm⟩ },
obtain ⟨t, ht, rfl⟩ | ⟨t, ht, rfl⟩ := sum.is_connected_iff.1 ⟨h, hs⟩,
{ exact or.inl ⟨t, ht.is_preconnected, rfl⟩ },
{ exact or.inr ⟨t, ht.is_preconnected, rf... | lemma | sum.is_preconnected_iff | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected",
"is_preconnected_empty",
"set.image_empty",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
connected_component (x : α) : set α | ⋃₀ { s : set α | is_preconnected s ∧ x ∈ s } | def | connected_component | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected"
] | The connected component of a point is the maximal connected set
that contains this point. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
connected_component_in (F : set α) (x : α) : set α | if h : x ∈ F then coe '' (connected_component (⟨x, h⟩ : F)) else ∅ | def | connected_component_in | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component"
] | 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. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
connected_component_in_eq_image {F : set α} {x : α} (h : x ∈ F) :
connected_component_in F x = coe '' (connected_component (⟨x, h⟩ : F)) | dif_pos h | lemma | connected_component_in_eq_image | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"connected_component_in"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
connected_component_in_eq_empty {F : set α} {x : α} (h : x ∉ F) :
connected_component_in F x = ∅ | dif_neg h | lemma | connected_component_in_eq_empty | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component_in"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_connected_component {x : α} : x ∈ connected_component x | mem_sUnion_of_mem (mem_singleton x) ⟨is_connected_singleton.is_preconnected, mem_singleton x⟩ | theorem | mem_connected_component | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
mem_connected_component_in {x : α} {F : set α} (hx : x ∈ F) :
x ∈ connected_component_in F x | by simp [connected_component_in_eq_image hx, mem_connected_component, hx] | theorem | mem_connected_component_in | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component_in",
"connected_component_in_eq_image",
"mem_connected_component"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
connected_component_nonempty {x : α} :
(connected_component x).nonempty | ⟨x, mem_connected_component⟩ | theorem | connected_component_nonempty | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
connected_component_in_nonempty_iff {x : α} {F : set α} :
(connected_component_in F x).nonempty ↔ x ∈ F | by { rw [connected_component_in], split_ifs; simp [connected_component_nonempty, h] } | theorem | connected_component_in_nonempty_iff | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component_in",
"connected_component_nonempty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
connected_component_in_subset (F : set α) (x : α) :
connected_component_in F x ⊆ F | by { rw [connected_component_in], split_ifs; simp } | theorem | connected_component_in_subset | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component_in"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_connected_component {x : α} : is_preconnected (connected_component x) | is_preconnected_sUnion x _ (λ _, and.right) (λ _, and.left) | theorem | is_preconnected_connected_component | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"is_preconnected",
"is_preconnected_sUnion"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_connected_component_in {x : α} {F : set α} :
is_preconnected (connected_component_in F x) | begin
rw [connected_component_in], split_ifs,
{ exact embedding_subtype_coe.to_inducing.is_preconnected_image.mpr
is_preconnected_connected_component },
{ exact is_preconnected_empty },
end | lemma | is_preconnected_connected_component_in | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component_in",
"is_preconnected",
"is_preconnected_connected_component",
"is_preconnected_empty"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected_connected_component {x : α} : is_connected (connected_component x) | ⟨⟨x, mem_connected_component⟩, is_preconnected_connected_component⟩ | theorem | is_connected_connected_component | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"is_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected_connected_component_in_iff {x : α} {F : set α} :
is_connected (connected_component_in F x) ↔ x ∈ F | by simp_rw [← connected_component_in_nonempty_iff, is_connected,
is_preconnected_connected_component_in, and_true] | lemma | is_connected_connected_component_in_iff | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component_in",
"connected_component_in_nonempty_iff",
"is_connected",
"is_preconnected_connected_component_in"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.subset_connected_component {x : α} {s : set α}
(H1 : is_preconnected s) (H2 : x ∈ s) : s ⊆ connected_component x | λ z hz, mem_sUnion_of_mem hz ⟨H1, H2⟩ | theorem | is_preconnected.subset_connected_component | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"is_preconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.subset_connected_component_in {x : α} {F : set α} (hs : is_preconnected s)
(hxs : x ∈ s) (hsF : s ⊆ F) : s ⊆ connected_component_in F x | begin
have : is_preconnected ((coe : F → α) ⁻¹' s),
{ refine embedding_subtype_coe.to_inducing.is_preconnected_image.mp _,
rwa [subtype.image_preimage_coe, inter_eq_left_iff_subset.mpr hsF] },
have h2xs : (⟨x, hsF hxs⟩ : F) ∈ coe ⁻¹' s := by { rw [mem_preimage], exact hxs },
have := this.subset_connected_co... | lemma | is_preconnected.subset_connected_component_in | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component_in",
"connected_component_in_eq_image",
"is_preconnected",
"subtype.image_preimage_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected.subset_connected_component {x : α} {s : set α}
(H1 : is_connected s) (H2 : x ∈ s) : s ⊆ connected_component x | H1.2.subset_connected_component H2 | theorem | is_connected.subset_connected_component | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"is_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected.connected_component_in {x : α} {F : set α} (h : is_preconnected F)
(hx : x ∈ F) : connected_component_in F x = F | (connected_component_in_subset F x).antisymm (h.subset_connected_component_in hx subset_rfl) | lemma | is_preconnected.connected_component_in | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component_in",
"connected_component_in_subset",
"is_preconnected",
"subset_rfl"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
connected_component_eq {x y : α} (h : y ∈ connected_component x) :
connected_component x = connected_component y | eq_of_subset_of_subset
(is_connected_connected_component.subset_connected_component h)
(is_connected_connected_component.subset_connected_component
(set.mem_of_mem_of_subset mem_connected_component
(is_connected_connected_component.subset_connected_component h))) | theorem | connected_component_eq | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"mem_connected_component",
"set.mem_of_mem_of_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
connected_component_eq_iff_mem {x y : α} :
connected_component x = connected_component y ↔ x ∈ connected_component y | ⟨λ h, h ▸ mem_connected_component, λ h, (connected_component_eq h).symm⟩ | theorem | connected_component_eq_iff_mem | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"connected_component_eq",
"mem_connected_component"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
connected_component_in_eq {x y : α} {F : set α} (h : y ∈ connected_component_in F x) :
connected_component_in F x = connected_component_in F y | begin
have hx : x ∈ F := connected_component_in_nonempty_iff.mp ⟨y, h⟩,
simp_rw [connected_component_in_eq_image hx] at h ⊢,
obtain ⟨⟨y, hy⟩, h2y, rfl⟩ := h,
simp_rw [subtype.coe_mk, connected_component_in_eq_image hy, connected_component_eq h2y]
end | lemma | connected_component_in_eq | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component_eq",
"connected_component_in",
"connected_component_in_eq_image",
"subtype.coe_mk"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
connected_component_in_univ (x : α) :
connected_component_in univ x = connected_component x | subset_antisymm
(is_preconnected_connected_component_in.subset_connected_component $
mem_connected_component_in trivial)
(is_preconnected_connected_component.subset_connected_component_in mem_connected_component $
subset_univ _) | theorem | connected_component_in_univ | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"connected_component_in",
"mem_connected_component",
"mem_connected_component_in",
"subset_antisymm"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
connected_component_disjoint {x y : α} (h : connected_component x ≠ connected_component y) :
disjoint (connected_component x) (connected_component y) | set.disjoint_left.2 (λ a h1 h2, h
((connected_component_eq h1).trans (connected_component_eq h2).symm)) | lemma | connected_component_disjoint | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"connected_component_eq",
"disjoint"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_closed_connected_component {x : α} :
is_closed (connected_component x) | closure_subset_iff_is_closed.1 $
is_connected_connected_component.closure.subset_connected_component $
subset_closure mem_connected_component | theorem | is_closed_connected_component | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"is_closed",
"mem_connected_component",
"subset_closure"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.image_connected_component_subset [topological_space β] {f : α → β}
(h : continuous f) (a : α) : f '' connected_component a ⊆ connected_component (f a) | (is_connected_connected_component.image f h.continuous_on).subset_connected_component
((mem_image f (connected_component a) (f a)).2 ⟨a, mem_connected_component, rfl⟩) | lemma | continuous.image_connected_component_subset | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"continuous",
"mem_connected_component",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.maps_to_connected_component [topological_space β] {f : α → β}
(h : continuous f) (a : α) : maps_to f (connected_component a) (connected_component (f a)) | maps_to'.2 $ h.image_connected_component_subset a | lemma | continuous.maps_to_connected_component | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"continuous",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_component_subset_connected_component {x : α} :
irreducible_component x ⊆ connected_component x | is_irreducible_irreducible_component.is_connected.subset_connected_component
mem_irreducible_component | theorem | irreducible_component_subset_connected_component | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"irreducible_component",
"mem_irreducible_component"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
connected_component_in_mono (x : α) {F G : set α} (h : F ⊆ G) :
connected_component_in F x ⊆ connected_component_in G x | begin
by_cases hx : x ∈ F,
{ rw [connected_component_in_eq_image hx, connected_component_in_eq_image (h hx),
← show (coe : G → α) ∘ inclusion h = coe, by ext ; refl, image_comp],
exact image_subset coe ((continuous_inclusion h).image_connected_component_subset ⟨x, hx⟩) },
{ rw connected_component_in_e... | lemma | connected_component_in_mono | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component_in",
"connected_component_in_eq_empty",
"connected_component_in_eq_image",
"continuous_inclusion",
"set.empty_subset"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preconnected_space (α : Type u) [topological_space α] : Prop | (is_preconnected_univ : is_preconnected (univ : set α)) | class | preconnected_space | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected",
"topological_space"
] | A preconnected space is one where there is no non-trivial open partition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
connected_space (α : Type u) [topological_space α] extends preconnected_space α : Prop | (to_nonempty : nonempty α) | class | connected_space | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"preconnected_space",
"topological_space"
] | A connected space is a nonempty one where there is no non-trivial open partition. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_connected_univ [connected_space α] : is_connected (univ : set α) | ⟨univ_nonempty, is_preconnected_univ⟩ | lemma | is_connected_univ | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_space",
"is_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_range [topological_space β] [preconnected_space α] {f : α → β}
(h : continuous f) : is_preconnected (range f) | @image_univ _ _ f ▸ is_preconnected_univ.image _ h.continuous_on | lemma | is_preconnected_range | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"continuous",
"is_preconnected",
"preconnected_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected_range [topological_space β] [connected_space α] {f : α → β} (h : continuous f) :
is_connected (range f) | ⟨range_nonempty f, is_preconnected_range h⟩ | lemma | is_connected_range | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_space",
"continuous",
"is_connected",
"is_preconnected_range",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
dense_range.preconnected_space [topological_space β] [preconnected_space α] {f : α → β}
(hf : dense_range f) (hc : continuous f) :
preconnected_space β | ⟨hf.closure_eq ▸ (is_preconnected_range hc).closure⟩ | lemma | dense_range.preconnected_space | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"continuous",
"dense_range",
"is_preconnected_range",
"preconnected_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
connected_space_iff_connected_component :
connected_space α ↔ ∃ x : α, connected_component x = univ | begin
split,
{ rintro ⟨⟨x⟩⟩,
exactI ⟨x, eq_univ_of_univ_subset $
is_preconnected_univ.subset_connected_component (mem_univ x)⟩ },
{ rintros ⟨x, h⟩,
haveI : preconnected_space α := ⟨by { rw ← h, exact is_preconnected_connected_component }⟩,
exact ⟨⟨x⟩⟩ }
end | lemma | connected_space_iff_connected_component | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"connected_space",
"is_preconnected_connected_component",
"preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preconnected_space_iff_connected_component :
preconnected_space α ↔ ∀ x : α, connected_component x = univ | begin
split,
{ intros h x,
exactI (eq_univ_of_univ_subset $
is_preconnected_univ.subset_connected_component (mem_univ x)) },
{ intros h,
casesI is_empty_or_nonempty α with hα hα,
{ exact ⟨by { rw (univ_eq_empty_iff.mpr hα), exact is_preconnected_empty }⟩ },
{ exact ⟨by { rw ← h (classical.ch... | lemma | preconnected_space_iff_connected_component | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"is_empty_or_nonempty",
"is_preconnected_connected_component",
"is_preconnected_empty",
"preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preconnected_space.connected_component_eq_univ {X : Type*} [topological_space X]
[h : preconnected_space X] (x : X) : connected_component x = univ | preconnected_space_iff_connected_component.mp h x | lemma | preconnected_space.connected_component_eq_univ | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"preconnected_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
preirreducible_space.preconnected_space (α : Type u) [topological_space α]
[preirreducible_space α] : preconnected_space α | ⟨(preirreducible_space.is_preirreducible_univ α).is_preconnected⟩ | instance | preirreducible_space.preconnected_space | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"preconnected_space",
"preirreducible_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
irreducible_space.connected_space (α : Type u) [topological_space α]
[irreducible_space α] : connected_space α | { to_nonempty := irreducible_space.to_nonempty α } | instance | irreducible_space.connected_space | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_space",
"irreducible_space",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_inter [preconnected_space α] {s t : set α} :
is_open s → is_open t → s ∪ t = univ → s.nonempty → t.nonempty → (s ∩ t).nonempty | by simpa only [univ_inter, univ_subset_iff] using
@preconnected_space.is_preconnected_univ α _ _ s t | theorem | nonempty_inter | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_open",
"preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen_iff [preconnected_space α] {s : set α} : is_clopen s ↔ s = ∅ ∨ s = univ | ⟨λ hs, classical.by_contradiction $ λ h,
have h1 : s ≠ ∅ ∧ sᶜ ≠ ∅, from ⟨mt or.inl h,
mt (λ h2, or.inr $ (by rw [← compl_compl s, h2, compl_empty] : s = univ)) h⟩,
let ⟨_, h2, h3⟩ := nonempty_inter hs.1 hs.2.is_open_compl (union_compl_self s)
(nonempty_iff_ne_empty.2 h1.1) (nonempty_iff_ne_empty.2 h1.2) in
... | theorem | is_clopen_iff | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"compl_compl",
"is_clopen",
"is_clopen_empty",
"is_clopen_univ",
"nonempty_inter",
"preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen.eq_univ [preconnected_space α] {s : set α} (h' : is_clopen s) (h : s.nonempty) :
s = univ | (is_clopen_iff.mp h').resolve_left h.ne_empty | lemma | is_clopen.eq_univ | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_clopen",
"preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
frontier_eq_empty_iff [preconnected_space α] {s : set α} :
frontier s = ∅ ↔ s = ∅ ∨ s = univ | is_clopen_iff_frontier_eq_empty.symm.trans is_clopen_iff | lemma | frontier_eq_empty_iff | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"frontier",
"is_clopen_iff",
"preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
nonempty_frontier_iff [preconnected_space α] {s : set α} :
(frontier s).nonempty ↔ s.nonempty ∧ s ≠ univ | by simp only [nonempty_iff_ne_empty, ne.def, frontier_eq_empty_iff, not_or_distrib] | lemma | nonempty_frontier_iff | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"frontier",
"frontier_eq_empty_iff",
"not_or_distrib",
"preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype.preconnected_space {s : set α} (h : is_preconnected s) :
preconnected_space s | { is_preconnected_univ := by rwa [← embedding_subtype_coe.to_inducing.is_preconnected_image,
image_univ, subtype.range_coe] } | lemma | subtype.preconnected_space | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected",
"preconnected_space",
"subtype.range_coe"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype.connected_space {s : set α} (h : is_connected s) :
connected_space s | { to_preconnected_space := subtype.preconnected_space h.is_preconnected,
to_nonempty := h.nonempty.to_subtype } | lemma | subtype.connected_space | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_space",
"is_connected",
"subtype.preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_iff_preconnected_space {s : set α} :
is_preconnected s ↔ preconnected_space s | ⟨subtype.preconnected_space,
begin
introI,
simpa using is_preconnected_univ.image (coe : s → α) continuous_subtype_coe.continuous_on
end⟩ | lemma | is_preconnected_iff_preconnected_space | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected",
"preconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_connected_iff_connected_space {s : set α} : is_connected s ↔ connected_space s | ⟨subtype.connected_space,
λ h, ⟨nonempty_subtype.mp h.2, is_preconnected_iff_preconnected_space.mpr h.1⟩⟩ | lemma | is_connected_iff_connected_space | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_space",
"is_connected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_preconnected_iff_subset_of_disjoint {s : set α} :
is_preconnected s ↔
∀ (u v : set α) (hu : is_open u) (hv : is_open v) (hs : s ⊆ u ∪ v) (huv : s ∩ (u ∩ v) = ∅),
s ⊆ u ∨ s ⊆ v | begin
split; intro h,
{ intros u v hu hv hs huv,
specialize h u v hu hv hs,
contrapose! huv,
rw ←nonempty_iff_ne_empty,
simp [not_subset] at huv,
rcases huv with ⟨⟨x, hxs, hxu⟩, ⟨y, hys, hyv⟩⟩,
have hxv : x ∈ v := or_iff_not_imp_left.mp (hs hxs) hxu,
have hyu : y ∈ u := or_iff_not_imp_ri... | lemma | is_preconnected_iff_subset_of_disjoint | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_open",
"is_preconnected"
] | A set `s` is preconnected if and only if
for every cover by two open sets that are disjoint on `s`,
it is contained in one of the two covering sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_connected_iff_sUnion_disjoint_open {s : set α} :
is_connected s ↔
∀ (U : finset (set α)) (H : ∀ (u v : set α), u ∈ U → v ∈ U → (s ∩ (u ∩ v)).nonempty → u = v)
(hU : ∀ u ∈ U, is_open u) (hs : s ⊆ ⋃₀ ↑U),
∃ u ∈ U, s ⊆ u | begin
rw [is_connected, is_preconnected_iff_subset_of_disjoint],
split; intro h,
{ intro U, apply finset.induction_on U,
{ rcases h.left,
suffices : s ⊆ ∅ → false, { simpa },
intro, solve_by_elim },
{ intros u U hu IH hs hU H,
rw [finset.coe_insert, sUnion_insert] at H,
cases h.2 u... | lemma | is_connected_iff_sUnion_disjoint_open | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"by_contradiction",
"finset",
"finset.coe_insert",
"finset.induction_on",
"finset.mem_insert",
"finset.mem_insert_of_mem",
"finset.mem_insert_self",
"finset.mem_singleton",
"is_connected",
"is_open",
"is_open_sUnion",
"is_preconnected_iff_subset_of_disjoint"
] | A set `s` is connected if and only if
for every cover by a finite collection of open sets that are pairwise disjoint on `s`,
it is contained in one of the members of the collection. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected.subset_clopen {s t : set α} (hs : is_preconnected s) (ht : is_clopen t)
(hne : (s ∩ t).nonempty) : s ⊆ t | begin
by_contra h,
have : (s ∩ tᶜ).nonempty := inter_compl_nonempty_iff.2 h,
obtain ⟨x, -, hx, hx'⟩ : (s ∩ (t ∩ tᶜ)).nonempty,
from hs t tᶜ ht.is_open ht.compl.is_open (λ x hx, em _) hne this,
exact hx' hx
end | theorem | is_preconnected.subset_clopen | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"by_contra",
"em",
"is_clopen",
"is_preconnected"
] | Preconnected sets are either contained in or disjoint to any given clopen set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
disjoint_or_subset_of_clopen {s t : set α} (hs : is_preconnected s) (ht : is_clopen t) :
disjoint s t ∨ s ⊆ t | (disjoint_or_nonempty_inter s t).imp_right $ hs.subset_clopen ht | theorem | disjoint_or_subset_of_clopen | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"disjoint",
"is_clopen",
"is_preconnected"
] | Preconnected sets are either contained in or disjoint to any given clopen set. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected_iff_subset_of_disjoint_closed :
is_preconnected s ↔
∀ (u v : set α) (hu : is_closed u) (hv : is_closed v) (hs : s ⊆ u ∪ v) (huv : s ∩ (u ∩ v) = ∅),
s ⊆ u ∨ s ⊆ v | begin
split; intro h,
{ intros u v hu hv hs huv,
rw is_preconnected_closed_iff at h,
specialize h u v hu hv hs,
contrapose! huv,
rw ←nonempty_iff_ne_empty,
simp [not_subset] at huv,
rcases huv with ⟨⟨x, hxs, hxu⟩, ⟨y, hys, hyv⟩⟩,
have hxv : x ∈ v := or_iff_not_imp_left.mp (hs hxs) hxu,
... | theorem | is_preconnected_iff_subset_of_disjoint_closed | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_closed",
"is_preconnected",
"is_preconnected_closed_iff"
] | A set `s` is preconnected if and only if
for every cover by two closed sets that are disjoint on `s`,
it is contained in one of the two covering sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected_iff_subset_of_fully_disjoint_closed {s : set α} (hs : is_closed s) :
is_preconnected s ↔
∀ (u v : set α) (hu : is_closed u) (hv : is_closed v) (hss : s ⊆ u ∪ v) (huv : disjoint u v),
s ⊆ u ∨ s ⊆ v | begin
split,
{ intros h u v hu hv hss huv,
apply is_preconnected_iff_subset_of_disjoint_closed.1 h u v hu hv hss,
rw [huv.inter_eq, inter_empty] },
intro H,
rw is_preconnected_iff_subset_of_disjoint_closed,
intros u v hu hv hss huv,
have H1 := H (u ∩ s) (v ∩ s),
rw [subset_inter_iff, subset_inter_... | theorem | is_preconnected_iff_subset_of_fully_disjoint_closed | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"disjoint",
"is_closed",
"is_closed.inter",
"is_preconnected",
"is_preconnected_iff_subset_of_disjoint_closed"
] | A closed set `s` is preconnected if and only if
for every cover by two closed sets that are disjoint,
it is contained in one of the two covering sets. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_clopen.connected_component_subset {x} (hs : is_clopen s) (hx : x ∈ s) :
connected_component x ⊆ s | is_preconnected_connected_component.subset_clopen hs ⟨x, mem_connected_component, hx⟩ | lemma | is_clopen.connected_component_subset | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"is_clopen",
"mem_connected_component"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
connected_component_subset_Inter_clopen {x : α} :
connected_component x ⊆ ⋂ Z : {Z : set α // is_clopen Z ∧ x ∈ Z}, Z | subset_Inter $ λ Z, Z.2.1.connected_component_subset Z.2.2 | lemma | connected_component_subset_Inter_clopen | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"is_clopen"
] | The connected component of a point is always a subset of the intersection of all its clopen
neighbourhoods. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_clopen.bUnion_connected_component_eq {Z : set α} (h : is_clopen Z) :
(⋃ x ∈ Z, connected_component x) = Z | subset.antisymm (Union₂_subset $ λ x, h.connected_component_subset) $
λ x hx, mem_Union₂_of_mem hx mem_connected_component | lemma | is_clopen.bUnion_connected_component_eq | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"is_clopen",
"mem_connected_component"
] | A clopen set is the union of its connected components. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
preimage_connected_component_connected [topological_space β] {f : α → β}
(connected_fibers : ∀ t : β, is_connected (f ⁻¹' {t}))
(hcl : ∀ (T : set β), is_closed T ↔ is_closed (f ⁻¹' T)) (t : β) :
is_connected (f ⁻¹' connected_component t) | begin
-- The following proof is essentially https://stacks.math.columbia.edu/tag/0377
-- although the statement is slightly different
have hf : surjective f := surjective.of_comp (λ t : β, (connected_fibers t).1),
split,
{ cases hf t with s hs,
use s,
rw [mem_preimage, hs],
exact mem_connected_co... | lemma | preimage_connected_component_connected | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"disjoint",
"disjoint.of_preimage",
"is_closed",
"is_closed.inter",
"is_closed_connected_component",
"is_connected",
"is_preconnected_connected_component",
"is_preconnected_iff_subset_of_fully_disjoint_closed",
"mem_connected_component",
"topological_space"
] | The preimage of a connected component is preconnected if the function has connected fibers
and a subset is closed iff the preimage is. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
quotient_map.preimage_connected_component [topological_space β] {f : α → β}
(hf : quotient_map f) (h_fibers : ∀ y : β, is_connected (f ⁻¹' {y})) (a : α) :
f ⁻¹' connected_component (f a) = connected_component a | ((preimage_connected_component_connected h_fibers
(λ _, hf.is_closed_preimage.symm) _).subset_connected_component mem_connected_component).antisymm
(hf.continuous.maps_to_connected_component a) | lemma | quotient_map.preimage_connected_component | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"is_connected",
"mem_connected_component",
"preimage_connected_component_connected",
"quotient_map",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
quotient_map.image_connected_component [topological_space β] {f : α → β}
(hf : quotient_map f) (h_fibers : ∀ y : β, is_connected (f ⁻¹' {y})) (a : α) :
f '' connected_component a = connected_component (f a) | by rw [← hf.preimage_connected_component h_fibers, image_preimage_eq _ hf.surjective] | lemma | quotient_map.image_connected_component | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"is_connected",
"quotient_map",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_connected_space (α : Type*) [topological_space α] : Prop | (open_connected_basis : ∀ x, (𝓝 x).has_basis (λ s : set α, is_open s ∧ x ∈ s ∧ is_connected s) id) | class | locally_connected_space | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_connected",
"is_open",
"topological_space"
] | A topological space is **locally connected** if each neighborhood filter admits a basis
of connected *open* sets. Note that it is equivalent to each point having a basis of connected
(non necessarily open) sets but in a non-trivial way, so we choose this definition and prove the
equivalence later in `locally_connected_... | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
locally_connected_space_iff_open_connected_basis : locally_connected_space α ↔
∀ x, (𝓝 x).has_basis (λ s : set α, is_open s ∧ x ∈ s ∧ is_connected s) id | ⟨@locally_connected_space.open_connected_basis _ _, locally_connected_space.mk⟩ | lemma | locally_connected_space_iff_open_connected_basis | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_connected",
"is_open",
"locally_connected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_connected_space_iff_open_connected_subsets :
locally_connected_space α ↔ ∀ (x : α) (U ∈ 𝓝 x), ∃ V ⊆ U, is_open V ∧ x ∈ V ∧ is_connected V | begin
rw locally_connected_space_iff_open_connected_basis,
congrm ∀ x, (_ : Prop),
split,
{ intros h U hU,
rcases h.mem_iff.mp hU with ⟨V, hV, hVU⟩,
exact ⟨V, hVU, hV⟩ },
{ exact λ h, ⟨λ U, ⟨λ hU, let ⟨V, hVU, hV⟩ := h U hU in ⟨V, hV, hVU⟩,
λ ⟨V, ⟨hV, hxV, _⟩, hVU⟩, mem_nhds_iff.mpr ⟨V... | lemma | locally_connected_space_iff_open_connected_subsets | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_connected",
"is_open",
"locally_connected_space",
"locally_connected_space_iff_open_connected_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
discrete_topology.to_locally_connected_space (α) [topological_space α]
[discrete_topology α] : locally_connected_space α | locally_connected_space_iff_open_connected_subsets.2 $ λ x _U hU,
⟨{x}, singleton_subset_iff.2 $ mem_of_mem_nhds hU, is_open_discrete _, mem_singleton _,
is_connected_singleton⟩ | instance | discrete_topology.to_locally_connected_space | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"discrete_topology",
"is_open_discrete",
"locally_connected_space",
"mem_of_mem_nhds",
"topological_space"
] | A space with discrete topology is a locally connected space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
connected_component_in_mem_nhds [locally_connected_space α] {F : set α} {x : α}
(h : F ∈ 𝓝 x) :
connected_component_in F x ∈ 𝓝 x | begin
rw (locally_connected_space.open_connected_basis x).mem_iff at h,
rcases h with ⟨s, ⟨h1s, hxs, h2s⟩, hsF⟩,
exact mem_nhds_iff.mpr ⟨s, h2s.is_preconnected.subset_connected_component_in hxs hsF, h1s, hxs⟩
end | lemma | connected_component_in_mem_nhds | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component_in",
"locally_connected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open.connected_component_in [locally_connected_space α] {F : set α} {x : α}
(hF : is_open F) :
is_open (connected_component_in F x) | begin
rw [is_open_iff_mem_nhds],
intros y hy,
rw [connected_component_in_eq hy],
exact connected_component_in_mem_nhds (is_open_iff_mem_nhds.mp hF y $
connected_component_in_subset F x hy)
end | lemma | is_open.connected_component_in | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component_in",
"connected_component_in_eq",
"connected_component_in_mem_nhds",
"connected_component_in_subset",
"is_open",
"is_open_iff_mem_nhds",
"locally_connected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_open_connected_component [locally_connected_space α] {x : α} :
is_open (connected_component x) | begin
rw ← connected_component_in_univ,
exact is_open_univ.connected_component_in
end | lemma | is_open_connected_component | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"connected_component_in_univ",
"is_open",
"locally_connected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_clopen_connected_component [locally_connected_space α] {x : α} :
is_clopen (connected_component x) | ⟨is_open_connected_component, is_closed_connected_component⟩ | lemma | is_clopen_connected_component | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"is_clopen",
"locally_connected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_connected_space_iff_connected_component_in_open :
locally_connected_space α ↔ ∀ F : set α, is_open F → ∀ x ∈ F,
is_open (connected_component_in F x) | begin
split,
{ introI h,
exact λ F hF x _, hF.connected_component_in },
{ intro h,
rw locally_connected_space_iff_open_connected_subsets,
refine (λ x U hU, ⟨connected_component_in (interior U) x,
(connected_component_in_subset _ _).trans interior_subset, h _ is_open_interior x _,
mem_conne... | lemma | locally_connected_space_iff_connected_component_in_open | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component_in",
"connected_component_in_subset",
"interior",
"interior_subset",
"is_open",
"is_open_interior",
"locally_connected_space",
"locally_connected_space_iff_open_connected_subsets",
"mem_connected_component_in"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_connected_space_iff_connected_subsets :
locally_connected_space α ↔ ∀ (x : α) (U ∈ 𝓝 x), ∃ V ∈ 𝓝 x, is_preconnected V ∧ V ⊆ U | begin
split,
{ rw locally_connected_space_iff_open_connected_subsets,
intros h x U hxU,
rcases h x U hxU with ⟨V, hVU, hV₁, hxV, hV₂⟩,
exact ⟨V, hV₁.mem_nhds hxV, hV₂.is_preconnected, hVU⟩ },
{ rw locally_connected_space_iff_connected_component_in_open,
refine λ h U hU x hxU, is_open_iff_mem_nhds.... | lemma | locally_connected_space_iff_connected_subsets | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component_in_eq",
"connected_component_in_subset",
"filter.mem_of_superset",
"is_preconnected",
"locally_connected_space",
"locally_connected_space_iff_connected_component_in_open",
"locally_connected_space_iff_open_connected_subsets",
"mem_of_mem_nhds"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_connected_space_iff_connected_basis :
locally_connected_space α ↔
∀ x, (𝓝 x).has_basis (λ s : set α, s ∈ 𝓝 x ∧ is_preconnected s) id | begin
rw locally_connected_space_iff_connected_subsets,
congrm ∀ x, (_ : Prop),
exact filter.has_basis_self.symm
end | lemma | locally_connected_space_iff_connected_basis | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected",
"locally_connected_space",
"locally_connected_space_iff_connected_subsets"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
locally_connected_space_of_connected_bases {ι : Type*} (b : α → ι → set α) (p : α → ι → Prop)
(hbasis : ∀ x, (𝓝 x).has_basis (p x) (b x))
(hconnected : ∀ x i, p x i → is_preconnected (b x i)) :
locally_connected_space α | begin
rw locally_connected_space_iff_connected_basis,
exact λ x, (hbasis x).to_has_basis
(λ i hi, ⟨b x i, ⟨(hbasis x).mem_of_mem hi, hconnected x i hi⟩, subset_rfl⟩)
(λ s hs, ⟨(hbasis x).index s hs.1,
⟨(hbasis x).property_index hs.1, (hbasis x).set_index_subset hs.1⟩⟩)
end | lemma | locally_connected_space_of_connected_bases | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected",
"locally_connected_space",
"locally_connected_space_iff_connected_basis"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_totally_disconnected (s : set α) : Prop | ∀ t, t ⊆ s → is_preconnected t → t.subsingleton | def | is_totally_disconnected | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected"
] | A set `s` is called totally disconnected if every subset `t ⊆ s` which is preconnected is
a subsingleton, ie either empty or a singleton. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_totally_disconnected_empty : is_totally_disconnected (∅ : set α) | λ _ ht _ _ x_in _ _, (ht x_in).elim | theorem | is_totally_disconnected_empty | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_totally_disconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_totally_disconnected_singleton {x} : is_totally_disconnected ({x} : set α) | λ _ ht _, subsingleton_singleton.anti ht | theorem | is_totally_disconnected_singleton | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_totally_disconnected"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totally_disconnected_space (α : Type u) [topological_space α] : Prop | (is_totally_disconnected_univ : is_totally_disconnected (univ : set α)) | class | totally_disconnected_space | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_totally_disconnected",
"topological_space"
] | A space is totally disconnected if all of its connected components are singletons. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_preconnected.subsingleton [totally_disconnected_space α]
{s : set α} (h : is_preconnected s) : s.subsingleton | totally_disconnected_space.is_totally_disconnected_univ s (subset_univ s) h | lemma | is_preconnected.subsingleton | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected",
"totally_disconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
pi.totally_disconnected_space {α : Type*} {β : α → Type*}
[t₂ : Πa, topological_space (β a)] [∀a, totally_disconnected_space (β a)] :
totally_disconnected_space (Π (a : α), β a) | ⟨λ t h1 h2,
have this : ∀ a, is_preconnected ((λ x : Π a, β a, x a) '' t),
from λ a, h2.image (λ x, x a) (continuous_apply a).continuous_on,
λ x x_in y y_in, funext $ λ a, (this a).subsingleton ⟨x, x_in, rfl⟩ ⟨y, y_in, rfl⟩⟩ | instance | pi.totally_disconnected_space | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"continuous_apply",
"continuous_on",
"is_preconnected",
"topological_space",
"totally_disconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
prod.totally_disconnected_space [topological_space β]
[totally_disconnected_space α] [totally_disconnected_space β] :
totally_disconnected_space (α × β) | ⟨λ t h1 h2,
have H1 : is_preconnected (prod.fst '' t), from h2.image prod.fst continuous_fst.continuous_on,
have H2 : is_preconnected (prod.snd '' t), from h2.image prod.snd continuous_snd.continuous_on,
λ x hx y hy, prod.ext
(H1.subsingleton ⟨x, hx, rfl⟩ ⟨y, hy, rfl⟩)
(H2.subsingleton ⟨x, hx, rfl⟩ ⟨y, hy, rfl⟩)⟩ | instance | prod.totally_disconnected_space | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_preconnected",
"prod.ext",
"topological_space",
"totally_disconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_totally_disconnected_of_clopen_set {X : Type*} [topological_space X]
(hX : ∀ {x y : X} (h_diff : x ≠ y), ∃ (U : set X) (h_clopen : is_clopen U), x ∈ U ∧ y ∉ U) :
is_totally_disconnected (set.univ : set X) | begin
rintro S - hS,
unfold set.subsingleton,
by_contra' h_contra,
rcases h_contra with ⟨x, hx, y, hy, hxy⟩,
obtain ⟨U, h_clopen, hxU, hyU⟩ := hX hxy,
specialize hS U Uᶜ h_clopen.1 h_clopen.compl.1 (λ a ha, em (a ∈ U)) ⟨x, hx, hxU⟩ ⟨y, hy, hyU⟩,
rw [inter_compl_self, set.inter_empty] at hS,
exact set.no... | lemma | is_totally_disconnected_of_clopen_set | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"em",
"is_clopen",
"is_totally_disconnected",
"set.inter_empty",
"set.not_nonempty_empty",
"set.subsingleton",
"topological_space"
] | Let `X` be a topological space, and suppose that for all distinct `x,y ∈ X`, there
is some clopen set `U` such that `x ∈ U` and `y ∉ U`. Then `X` is totally disconnected. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totally_disconnected_space_iff_connected_component_subsingleton :
totally_disconnected_space α ↔ ∀ x : α, (connected_component x).subsingleton | begin
split,
{ intros h x,
apply h.1,
{ exact subset_univ _ },
exact is_preconnected_connected_component },
intro h, constructor,
intros s s_sub hs,
rcases eq_empty_or_nonempty s with rfl | ⟨x, x_in⟩,
{ exact subsingleton_empty },
{ exact (h x).anti (hs.subset_connected_component x_in) }
end | lemma | totally_disconnected_space_iff_connected_component_subsingleton | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"is_preconnected_connected_component",
"totally_disconnected_space"
] | A space is totally disconnected iff its connected components are subsingletons. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totally_disconnected_space_iff_connected_component_singleton :
totally_disconnected_space α ↔ ∀ x : α, connected_component x = {x} | begin
rw totally_disconnected_space_iff_connected_component_subsingleton,
apply forall_congr (λ x, _),
rw subsingleton_iff_singleton,
exact mem_connected_component
end | lemma | totally_disconnected_space_iff_connected_component_singleton | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"mem_connected_component",
"totally_disconnected_space",
"totally_disconnected_space_iff_connected_component_subsingleton"
] | A space is totally disconnected iff its connected components are singletons. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
connected_component_eq_singleton [totally_disconnected_space α] (x : α) :
connected_component x = {x} | totally_disconnected_space_iff_connected_component_singleton.1 ‹_› x | theorem | connected_component_eq_singleton | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"totally_disconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
continuous.image_connected_component_eq_singleton {β : Type*} [topological_space β]
[totally_disconnected_space β] {f : α → β} (h : continuous f) (a : α) :
f '' connected_component a = {f a} | (set.subsingleton_iff_singleton $ mem_image_of_mem f mem_connected_component).mp
(is_preconnected_connected_component.image f h.continuous_on).subsingleton | lemma | continuous.image_connected_component_eq_singleton | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"continuous",
"mem_connected_component",
"set.subsingleton_iff_singleton",
"topological_space",
"totally_disconnected_space"
] | The image of a connected component in a totally disconnected space is a singleton. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_totally_disconnected_of_totally_disconnected_space [totally_disconnected_space α]
(s : set α) : is_totally_disconnected s | λ t hts ht, totally_disconnected_space.is_totally_disconnected_univ _ t.subset_univ ht | lemma | is_totally_disconnected_of_totally_disconnected_space | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_totally_disconnected",
"totally_disconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_totally_disconnected_of_image [topological_space β] {f : α → β} (hf : continuous_on f s)
(hf' : injective f) (h : is_totally_disconnected (f '' s)) : is_totally_disconnected s | λ t hts ht x x_in y y_in, hf' $ h _ (image_subset f hts) (ht.image f $ hf.mono hts)
(mem_image_of_mem f x_in) (mem_image_of_mem f y_in) | lemma | is_totally_disconnected_of_image | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"continuous_on",
"is_totally_disconnected",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
embedding.is_totally_disconnected [topological_space β] {f : α → β} (hf : embedding f)
{s : set α} (h : is_totally_disconnected (f '' s)) : is_totally_disconnected s | is_totally_disconnected_of_image hf.continuous.continuous_on hf.inj h | lemma | embedding.is_totally_disconnected | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"embedding",
"is_totally_disconnected",
"is_totally_disconnected_of_image",
"topological_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
subtype.totally_disconnected_space {α : Type*} {p : α → Prop} [topological_space α]
[totally_disconnected_space α] : totally_disconnected_space (subtype p) | ⟨embedding_subtype_coe.is_totally_disconnected
(is_totally_disconnected_of_totally_disconnected_space _)⟩ | instance | subtype.totally_disconnected_space | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_totally_disconnected_of_totally_disconnected_space",
"topological_space",
"totally_disconnected_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_totally_separated (s : set α) : Prop | ∀ x ∈ s, ∀ y ∈ s, x ≠ y → ∃ u v : set α, is_open u ∧ is_open v ∧
x ∈ u ∧ y ∈ v ∧ s ⊆ u ∪ v ∧ disjoint u v | def | is_totally_separated | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"disjoint",
"is_open"
] | A set `s` is called totally separated if any two points of this set can be separated
by two disjoint open sets covering `s`. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
is_totally_separated_empty : is_totally_separated (∅ : set α) | λ x, false.elim | theorem | is_totally_separated_empty | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_totally_separated"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_totally_separated_singleton {x} : is_totally_separated ({x} : set α) | λ p hp q hq hpq, (hpq $ (eq_of_mem_singleton hp).symm ▸ (eq_of_mem_singleton hq).symm).elim | theorem | is_totally_separated_singleton | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_totally_separated"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
is_totally_disconnected_of_is_totally_separated {s : set α}
(H : is_totally_separated s) : is_totally_disconnected s | begin
intros t hts ht x x_in y y_in,
by_contra h,
obtain ⟨u : set α, v : set α, hu : is_open u, hv : is_open v,
hxu : x ∈ u, hyv : y ∈ v, hs : s ⊆ u ∪ v, huv⟩ :=
H x (hts x_in) y (hts y_in) h,
refine (ht _ _ hu hv (hts.trans hs) ⟨x, x_in, hxu⟩ ⟨y, y_in, hyv⟩).ne_empty _,
rw [huv.inter_eq, inter_... | theorem | is_totally_disconnected_of_is_totally_separated | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"by_contra",
"is_open",
"is_totally_disconnected",
"is_totally_separated"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totally_separated_space (α : Type u) [topological_space α] : Prop | (is_totally_separated_univ [] : is_totally_separated (univ : set α)) | class | totally_separated_space | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_totally_separated",
"topological_space"
] | A space is totally separated if any two points can be separated by two disjoint open sets
covering the whole space. | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
totally_separated_space.totally_disconnected_space (α : Type u) [topological_space α]
[totally_separated_space α] : totally_disconnected_space α | ⟨is_totally_disconnected_of_is_totally_separated $
totally_separated_space.is_totally_separated_univ α⟩ | instance | totally_separated_space.totally_disconnected_space | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"topological_space",
"totally_disconnected_space",
"totally_separated_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
totally_separated_space.of_discrete
(α : Type*) [topological_space α] [discrete_topology α] : totally_separated_space α | ⟨λ a _ b _ h, ⟨{b}ᶜ, {b}, is_open_discrete _, is_open_discrete _, by simpa⟩⟩ | instance | totally_separated_space.of_discrete | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"discrete_topology",
"is_open_discrete",
"topological_space",
"totally_separated_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
exists_clopen_of_totally_separated {α : Type*} [topological_space α]
[totally_separated_space α] {x y : α} (hxy : x ≠ y) :
∃ (U : set α) (hU : is_clopen U), x ∈ U ∧ y ∈ Uᶜ | begin
obtain ⟨U, V, hU, hV, Ux, Vy, f, disj⟩ :=
totally_separated_space.is_totally_separated_univ α x (set.mem_univ x) y (set.mem_univ y) hxy,
have clopen_U := is_clopen_inter_of_disjoint_cover_clopen (is_clopen_univ) f hU hV disj,
rw univ_inter _ at clopen_U,
rw [←set.subset_compl_iff_disjoint_right, subse... | lemma | exists_clopen_of_totally_separated | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"is_clopen",
"is_clopen_inter_of_disjoint_cover_clopen",
"is_clopen_univ",
"set.mem_univ",
"topological_space",
"totally_separated_space"
] | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 | |
connected_component_setoid (α : Type*) [topological_space α] : setoid α | ⟨λ x y, connected_component x = connected_component y,
⟨λ x, by trivial, λ x y h1, h1.symm, λ x y z h1 h2, h1.trans h2⟩⟩ | def | connected_component_setoid | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component",
"topological_space"
] | The setoid of connected components of a topological space | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
connected_components (α : Type u) [topological_space α] | quotient (connected_component_setoid α) | def | connected_components | topology | src/topology/connected.lean | [
"data.set.bool_indicator",
"order.succ_pred.relation",
"topology.subset_properties",
"tactic.congrm"
] | [
"connected_component_setoid",
"topological_space"
] | The quotient of a space by its connected components | https://github.com/leanprover-community/mathlib | 65a1391a0106c9204fe45bc73a039f056558cb83 |
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