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lines changed Original file line number Diff line number Diff line change @@ -42,16 +42,20 @@ instance (priority := 100) DiscreteTopology.secondCountableTopology_of_countable
4242 (iUnion_of_singleton α)
4343
4444set_option backward.isDefEq.respectTransparency false in
45- theorem LinearOrder.bot_topologicalSpace_eq_generateFrom {α} [LinearOrder α] [PredOrder α]
46- [SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by
45+ theorem LinearOrder.bot_topologicalSpace_eq_preorderTopology {α} [LinearOrder α] [PredOrder α]
46+ [SuccOrder α] : (⊥ : TopologicalSpace α) = Preorder.topology α := by
4747 let _ := Preorder.topology α
4848 have : OrderTopology α := ⟨rfl⟩
4949 exact DiscreteTopology.of_predOrder_succOrder.eq_bot.symm
5050
51+ @ [deprecated (since := "2026-03-22" )]
52+ alias LinearOrder.bot_topologicalSpace_eq_generateFrom :=
53+ LinearOrder.bot_topologicalSpace_eq_preorderTopology
54+
5155theorem discreteTopology_iff_orderTopology_of_pred_succ [LinearOrder α] [PredOrder α]
5256 [SuccOrder α] : DiscreteTopology α ↔ OrderTopology α := by
5357 refine ⟨fun h ↦ ⟨?_⟩, fun h ↦ .of_predOrder_succOrder⟩
54- rw [h.eq_bot, LinearOrder.bot_topologicalSpace_eq_generateFrom ]
58+ rw [h.eq_bot, LinearOrder.bot_topologicalSpace_eq_preorderTopology ]
5559
5660instance OrderTopology.of_discreteTopology [LinearOrder α] [PredOrder α] [SuccOrder α]
5761 [DiscreteTopology α] : OrderTopology α :=
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