Instances related to the discrete topology

We prove that the discrete topology is

• first-countable,
• second-countable for an encodable type,
• equal to the order topology in linear orders which are also PredOrder and SuccOrder,
• metrizable.

When importing this file and Data.Nat.SuccPred, the instances SecondCountableTopology ℕ and OrderTopology ℕ become available.

@[instance 100]

instance DiscreteTopology.firstCountableTopology {α : Type u_1} [TopologicalSpace α] [DiscreteTopology α] : FirstCountableTopology α

@[instance 100]

instance DiscreteTopology.secondCountableTopology_of_countable {α : Type u_1} [TopologicalSpace α] [hd : DiscreteTopology α] [Countable α] : SecondCountableTopology α

@[deprecated DiscreteTopology.secondCountableTopology_of_countable]

theorem DiscreteTopology.secondCountableTopology_of_encodable {α : Type u_2} [TopologicalSpace α] [DiscreteTopology α] [Countable α] : SecondCountableTopology α

theorem LinearOrder.bot_topologicalSpace_eq_generateFrom {α : Type u_2} [LinearOrder α] [PredOrder α] [SuccOrder α] : ⊥ = TopologicalSpace.generateFrom {s : Set α | ∃ (a : α), s = Set.Ioi a ∨ s = Set.Iio a}

@[deprecated LinearOrder.bot_topologicalSpace_eq_generateFrom]

theorem bot_topologicalSpace_eq_generateFrom_of_pred_succOrder {α : Type u_2} [LinearOrder α] [PredOrder α] [SuccOrder α] : ⊥ = TopologicalSpace.generateFrom {s : Set α | ∃ (a : α), s = Set.Ioi a ∨ s = Set.Iio a}

Alias of LinearOrder.bot_topologicalSpace_eq_generateFrom.

theorem discreteTopology_iff_orderTopology_of_pred_succ {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [PredOrder α] [SuccOrder α] : DiscreteTopology α ↔ OrderTopology α

@[deprecated discreteTopology_iff_orderTopology_of_pred_succ]

theorem discreteTopology_iff_orderTopology_of_pred_succ' {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [PredOrder α] [SuccOrder α] : DiscreteTopology α ↔ OrderTopology α

Alias of discreteTopology_iff_orderTopology_of_pred_succ.

instance OrderTopology.of_discreteTopology {α : Type u_1} [TopologicalSpace α] [LinearOrder α] [PredOrder α] [SuccOrder α] [DiscreteTopology α] : OrderTopology α