@@ -114,12 +114,32 @@ open separated
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lemma comm (s t : set α) : separated s t ↔ separated t s :=
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⟨symm, symm⟩
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+ lemma preimage [topological_space β] {f : α → β} {s t : set β} (h : separated s t)
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+ (hf : continuous f) : separated (f ⁻¹' s) (f ⁻¹' t) :=
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+ let ⟨U, V, oU, oV, sU, tV, UV⟩ := h in
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+ ⟨f ⁻¹' U, f ⁻¹' V, oU.preimage hf, oV.preimage hf, preimage_mono sU, preimage_mono tV,
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+ UV.preimage f⟩
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+
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+ protected lemma disjoint {s t : set α} (h : separated s t) : disjoint s t :=
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+ let ⟨U, V, hU, hV, hsU, htV, hd⟩ := h in hd.mono hsU htV
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+
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+ lemma disjoint_closure_left {s t : set α} (h : separated s t) : disjoint (closure s) t :=
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+ let ⟨U, V, hU, hV, hsU, htV, hd⟩ := h
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+ in (hd.closure_left hV).mono (closure_mono hsU) htV
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+
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+ lemma disjoint_closure_right {s t : set α} (h : separated s t) : disjoint s (closure t) :=
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+ h.symm.disjoint_closure_left.symm
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+
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lemma empty_right (a : set α) : separated a ∅ :=
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⟨_, _, is_open_univ, is_open_empty, λ a h, mem_univ a, λ a h, by cases h, disjoint_empty _⟩
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lemma empty_left (a : set α) : separated ∅ a :=
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(empty_right _).symm
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+ lemma mono {s₁ s₂ t₁ t₂ : set α} (h : separated s₂ t₂) (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t₂) :
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+ separated s₁ t₁ :=
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+ let ⟨U, V, hU, hV, hsU, htV, hd⟩ := h in ⟨U, V, hU, hV, hs.trans hsU, ht.trans htV, hd⟩
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+
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lemma union_left {a b c : set α} : separated a c → separated b c → separated (a ∪ b) c :=
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λ ⟨U, V, oU, oV, aU, bV, UV⟩ ⟨W, X, oW, oX, aW, bX, WX⟩,
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⟨U ∪ W, V ∩ X, is_open.union oU oW, is_open.inter oV oX,
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