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[Merged by Bors] - chore(Topology/Instances/Discrete): merge 2 instances #11296

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11 changes: 6 additions & 5 deletions Mathlib/Topology/Instances/Discrete.lean
Original file line number Diff line number Diff line change
Expand Up @@ -33,18 +33,19 @@ instance (priority := 100) DiscreteTopology.firstCountableTopology [DiscreteTopo
nhds_generated_countable := by rw [nhds_discrete]; exact isCountablyGenerated_pure
#align discrete_topology.first_countable_topology DiscreteTopology.firstCountableTopology

instance (priority := 100) DiscreteTopology.secondCountableTopology_of_encodable
[hd : DiscreteTopology α] [Encodable α] : SecondCountableTopology α :=
instance (priority := 100) DiscreteTopology.secondCountableTopology_of_countable
[hd : DiscreteTopology α] [Countable α] : SecondCountableTopology α :=
haveI : ∀ i : α, SecondCountableTopology (↥({i} : Set α)) := fun i =>
{ is_open_generated_countable :=
⟨{univ}, countable_singleton _, by simp only [eq_iff_true_of_subsingleton]⟩ }
secondCountableTopology_of_countable_cover (singletons_open_iff_discrete.mpr hd)
(iUnion_of_singleton α)
#align discrete_topology.second_countable_topology_of_encodable DiscreteTopology.secondCountableTopology_of_encodable
#align discrete_topology.second_countable_topology_of_encodable DiscreteTopology.secondCountableTopology_of_countable

instance (priority := 100) DiscreteTopology.secondCountableTopology_of_countable {α : Type*}
@[deprecated DiscreteTopology.secondCountableTopology_of_countable] -- 2024-03-11
theorem DiscreteTopology.secondCountableTopology_of_encodable {α : Type*}
[TopologicalSpace α] [DiscreteTopology α] [Countable α] : SecondCountableTopology α :=
@DiscreteTopology.secondCountableTopology_of_encodable _ _ _ (Encodable.ofCountable _)
DiscreteTopology.secondCountableTopology_of_countable
#align discrete_topology.second_countable_topology_of_countable DiscreteTopology.secondCountableTopology_of_countable

theorem bot_topologicalSpace_eq_generateFrom_of_pred_succOrder [PartialOrder α] [PredOrder α]
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