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Mathlib/Topology/Coherent.lean

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@@ -92,24 +92,6 @@ lemma of_nhds (h : ∀ x, ∃ s ∈ S, s ∈ 𝓝 x) : IsCoherentWith S :=
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let ⟨s, hsS, hsx⟩ := h x
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(hf s hsS).continuousAt hsx
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lemma of_open_cover (hSO : ∀ s ∈ S, IsOpen s) (hSU : ⋃₀ S = univ) : IsCoherentWith S :=
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of_nhds (S := S) fun x ↦
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let ⟨s, hsS, hsx⟩ := sUnion_eq_univ_iff.mp hSU x
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⟨s, hsS, hSO _ hsS |>.mem_nhds hsx⟩
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-- lemma of_closed_cover (hSC : ∀ s ∈ S, IsClosed s) (hSU : ⋃ s ∈ S, sᶜ = univ) :
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-- IsCoherentWith S := by
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-- have hSc : IsCoherentWith { sᶜ | s ∈ S } := by
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-- apply of_open_cover <;> simpa [sUnion_eq_iUnion, iUnion_subtype]
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-- apply of_isClosed
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-- intro t ht
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-- simp_rw +contextual [(hSC _ _).inter_preimage_val_iff] at ht
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-- rw [hSc.isClosed_iff]
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-- simp
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-- intro s hs
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-- rw [← isOpen_compl_iff, ← preimage_compl, (hSC s hs).isOpen_compl.inter_preimage_val_iff,
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-- ← compl_union, isOpen_compl_iff]
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/-- If a space `X` is coherent with an indexed family of subspaces `S` whose union is `X`, then the
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canonical inclusion from `Σ i, S i` to `X` is a quotient map. -/
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lemma isQuotientMap_sigma_desc'

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