214 lines
7.3 KiB
Text
214 lines
7.3 KiB
Text
/-
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Copyright (c) 2015 Jacob Gross. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Jacob Gross, Jeremy Avigad
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Open and closed sets, seperation axioms and generated topologies.
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-/
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import data.set data.nat
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open algebra eq.ops set nat
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structure topology [class] (X : Type) :=
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(opens : set (set X))
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(univ_mem_opens : univ ∈ opens)
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(sUnion_mem_opens : ∀ {S : set (set X)}, S ⊆ opens → ⋃₀ S ∈ opens)
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(inter_mem_opens : ∀₀ s ∈ opens, ∀₀ t ∈ opens, s ∩ t ∈ opens)
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namespace topology
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variables {X : Type} [topology X]
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/- open sets -/
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definition Open (s : set X) : Prop := s ∈ opens X
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theorem Open_empty : Open (∅ : set X) :=
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have ∅ ⊆ opens X, from empty_subset _,
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have ⋃₀ ∅ ∈ opens X, from sUnion_mem_opens this,
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show ∅ ∈ opens X, using this, by rewrite -sUnion_empty; apply this
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theorem Open_univ : Open (univ : set X) :=
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univ_mem_opens X
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theorem Open_sUnion {S : set (set X)} (H : ∀₀ t ∈ S, Open t) : Open (⋃₀ S) :=
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sUnion_mem_opens H
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theorem Open_Union {I : Type} {s : I → set X} (H : ∀ i, Open (s i)) : Open (⋃ i, s i) :=
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have ∀₀ t ∈ s ' univ, Open t,
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from take t, suppose t ∈ s ' univ,
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obtain i [univi (Hi : s i = t)], from this,
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show Open t, by rewrite -Hi; exact H i,
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using this, by rewrite Union_eq_sUnion_image; apply Open_sUnion this
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theorem Open_union {s t : set X} (Hs : Open s) (Ht : Open t) : Open (s ∪ t) :=
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have ∀ i, Open (bin_ext s t i), by intro i; cases i; exact Hs; exact Ht,
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show Open (s ∪ t), using this, by rewrite -Union_bin_ext; exact Open_Union this
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theorem Open_inter {s t : set X} (Hs : Open s) (Ht : Open t) : Open (s ∩ t) :=
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inter_mem_opens X Hs Ht
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theorem Open_sInter_of_finite {s : set (set X)} [fins : finite s] (H : ∀₀ t ∈ s, Open t) :
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Open (⋂₀ s) :=
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begin
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induction fins with a s fins anins ih,
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{rewrite sInter_empty, exact Open_univ},
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rewrite sInter_insert,
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apply Open_inter,
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show Open a, from H (mem_insert a s),
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apply ih, intros t ts,
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show Open t, from H (mem_insert_of_mem a ts)
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end
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/- closed sets -/
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definition closed [reducible] (s : set X) : Prop := Open (-s)
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theorem closed_iff_Open_comp (s : set X) : closed s ↔ Open (-s) := !iff.refl
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theorem Open_iff_closed_comp (s : set X) : Open s ↔ closed (-s) :=
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by rewrite [closed_iff_Open_comp, comp_comp]
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theorem closed_comp {s : set X} (H : Open s) : closed (-s) :=
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by rewrite [-Open_iff_closed_comp]; apply H
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theorem closed_empty : closed (∅ : set X) :=
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by rewrite [↑closed, comp_empty]; exact Open_univ
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theorem closed_univ : closed (univ : set X) :=
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by rewrite [↑closed, comp_univ]; exact Open_empty
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theorem closed_sInter {S : set (set X)} (H : ∀₀ t ∈ S, closed t) : closed (⋂₀ S) :=
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begin
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rewrite [↑closed, comp_sInter],
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apply Open_sUnion,
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intro t,
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rewrite [mem_image_complement, Open_iff_closed_comp],
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apply H
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end
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theorem closed_Inter {I : Type} {s : I → set X} (H : ∀ i, closed (s i : set X)) :
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closed (⋂ i, s i) :=
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by rewrite [↑closed, comp_Inter]; apply Open_Union; apply H
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theorem closed_inter {s t : set X} (Hs : closed s) (Ht : closed t) : closed (s ∩ t) :=
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by rewrite [↑closed, comp_inter]; apply Open_union; apply Hs; apply Ht
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theorem closed_union {s t : set X} (Hs : closed s) (Ht : closed t) : closed (s ∪ t) :=
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by rewrite [↑closed, comp_union]; apply Open_inter; apply Hs; apply Ht
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theorem closed_sUnion_of_finite {s : set (set X)} [fins : finite s] (H : ∀₀ t ∈ s, closed t) :
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closed (⋂₀ s) :=
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begin
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rewrite [↑closed, comp_sInter],
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apply Open_sUnion,
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intro t,
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rewrite [mem_image_complement, Open_iff_closed_comp],
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apply H
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end
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theorem open_diff {s t : set X} (Hs : Open s) (Ht : closed t) : Open (s \ t) :=
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Open_inter Hs Ht
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theorem closed_diff {s t : set X} (Hs : closed s) (Ht : Open t) : closed (s \ t) :=
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closed_inter Hs (closed_comp Ht)
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end topology
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/- separation -/
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structure T0_space [class] (X : Type) extends topology X :=
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(T0 : ∀ {x y}, x ≠ y → ∃ U, U ∈ opens ∧ ¬(x ∈ U ↔ y ∈ U))
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namespace topology
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variables {X : Type} [T0_space X]
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theorem T0 {x y : X} (H : x ≠ y) : ∃ U, Open U ∧ ¬(x ∈ U ↔ y ∈ U) :=
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T0_space.T0 H
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end topology
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structure T1_space [class] (X : Type) extends topology X :=
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(T1 : ∀ {x y}, x ≠ y → ∃ U, U ∈ opens ∧ x ∈ U ∧ y ∉ U)
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protected definition T0_space.of_T1 [reducible] [trans_instance] {X : Type} [T : T1_space X] :
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T0_space X :=
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⦃T0_space, T,
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T0 := abstract
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take x y, assume H,
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obtain U [Uopens [xU ynU]], from T1_space.T1 H,
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exists.intro U (and.intro Uopens
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(show ¬ (x ∈ U ↔ y ∈ U), from assume H, ynU (iff.mp H xU)))
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end ⦄
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namespace topology
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variables {X : Type} [T1_space X]
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theorem T1 {x y : X} (H : x ≠ y) : ∃ U, Open U ∧ x ∈ U ∧ y ∉ U :=
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T1_space.T1 H
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end topology
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structure T2_space [class] (X : Type) extends topology X :=
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(T2 : ∀ {x y}, x ≠ y → ∃ U V, U ∈ opens ∧ V ∈ opens ∧ x ∈ U ∧ y ∈ V ∧ U ∩ V = ∅)
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protected definition T1_space.of_T2 [reducible] [trans_instance] {X : Type} [T : T2_space X] :
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T1_space X :=
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⦃T1_space, T,
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T1 := abstract
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take x y, assume H,
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obtain U [V [Uopens [Vopens [xU [yV UVempty]]]]], from T2_space.T2 H,
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exists.intro U (and.intro Uopens (and.intro xU
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(show y ∉ U, from assume yU,
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have y ∈ U ∩ V, from and.intro yU yV,
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show y ∈ ∅, from UVempty ▸ this)))
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end ⦄
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namespace topology
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variables {X : Type} [T2_space X]
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theorem T2 {x y : X} (H : x ≠ y) : ∃ U V, Open U ∧ Open V ∧ x ∈ U ∧ y ∈ V ∧ U ∩ V = ∅ :=
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T2_space.T2 H
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end topology
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structure perfect_space [class] (X : Type) extends topology X :=
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(perfect : ∀ x, '{x} ∉ opens)
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/- topology generated by a set -/
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namespace topology
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inductive opens_generated_by {X : Type} (B : set (set X)) : set X → Prop :=
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| generators_mem : ∀ ⦃s : set X⦄, s ∈ B → opens_generated_by B s
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| univ_mem : opens_generated_by B univ
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| inter_mem : ∀ ⦃s t⦄, opens_generated_by B s → opens_generated_by B t →
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opens_generated_by B (s ∩ t)
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| sUnion_mem : ∀ ⦃S : set (set X)⦄, S ⊆ opens_generated_by B → opens_generated_by B (⋃₀ S)
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protected definition generated_by [instance] [reducible] {X : Type} (B : set (set X)) : topology X :=
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⦃topology,
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opens := opens_generated_by B,
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univ_mem_opens := opens_generated_by.univ_mem B,
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inter_mem_opens := λ s Hs t Ht, opens_generated_by.inter_mem Hs Ht,
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sUnion_mem_opens := opens_generated_by.sUnion_mem
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⦄
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theorem generators_mem_topology_generated_by {X : Type} (B : set (set X)) :
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let T := topology.generated_by B in
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∀₀ s ∈ B, @Open _ T s :=
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λ s H, opens_generated_by.generators_mem H
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theorem opens_generated_by_initial {X : Type} {B : set (set X)} {T : topology X} (H : B ⊆ @opens _ T) :
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opens_generated_by B ⊆ @opens _ T :=
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begin
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intro s Hs,
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induction Hs with s sB s t os ot soX toX S SB SOX,
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{exact H sB},
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{exact univ_mem_opens X},
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{exact inter_mem_opens X soX toX},
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exact sUnion_mem_opens SOX
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end
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theorem topology_generated_by_initial {X : Type} {B : set (set X)} {T : topology X}
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(H : ∀₀ s ∈ B, @Open _ T s) {s : set X} (H1 : @Open _ (topology.generated_by B) s) :
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@Open _ T s :=
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opens_generated_by_initial H H1
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end topology
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