Specialization order

This file defines a type synonym for a topological space considered with its specialisation order.

def Specialization (α : Type u_1) : Type u_1

Type synonym for a topological space considered with its specialisation order.

Equations

• Specialization α = α

@[match_pattern]

def Specialization.toEquiv {α : Type u_1} : α ≃ Specialization α

toEquiv is the "identity" function to the Specialization of a type.

Equations

• Specialization.toEquiv = Equiv.refl α

@[match_pattern]

def Specialization.ofEquiv {α : Type u_1} : Specialization α ≃ α

ofEquiv is the identity function from the Specialization of a type.

Equations

• Specialization.ofEquiv = Equiv.refl (Specialization α)

@[simp]

theorem Specialization.toEquiv_symm {α : Type u_1} : toEquiv.symm = ofEquiv

@[simp]

theorem Specialization.ofEquiv_symm {α : Type u_1} : ofEquiv.symm = toEquiv

@[simp]

theorem Specialization.toEquiv_ofEquiv {α : Type u_1} (a : Specialization α) : toEquiv (ofEquiv a) = a

@[simp]

theorem Specialization.ofEquiv_toEquiv {α : Type u_1} (a : α) : ofEquiv (toEquiv a) = a

@[simp]

theorem Specialization.toEquiv_inj {α : Type u_1} {a b : α} : toEquiv a = toEquiv b ↔ a = b

@[simp]

theorem Specialization.ofEquiv_inj {α : Type u_1} {a b : Specialization α} : ofEquiv a = ofEquiv b ↔ a = b

def Specialization.rec {α : Type u_1} {β : Specialization α → Sort u_4} (h : (a : α) → β (toEquiv a)) (a : Specialization α) : β a

A recursor for Specialization. Use as induction x.

Equations

• Specialization.rec h a = h (Specialization.ofEquiv a)

@[instance_reducible]

instance Specialization.instPreorder {α : Type u_1} [TopologicalSpace α] : Preorder (Specialization α)

Equations

• Specialization.instPreorder = { le := fun (a a_1 : Specialization α) => a_1 ⤳ a, le_refl := ⋯, le_trans := ⋯, lt_iff_le_not_ge := ⋯ }

@[instance_reducible]

instance Specialization.instPartialOrder {α : Type u_1} [TopologicalSpace α] [T0Space α] : PartialOrder (Specialization α)

Equations

• Specialization.instPartialOrder = { toPreorder := Specialization.instPreorder, le_antisymm := ⋯ }

@[simp]

theorem Specialization.toEquiv_le_toEquiv {α : Type u_1} [TopologicalSpace α] {a b : α} : toEquiv a ≤ toEquiv b ↔ b ⤳ a

@[simp]

theorem Specialization.ofEquiv_specializes_ofEquiv {α : Type u_1} [TopologicalSpace α] {a b : Specialization α} : ofEquiv a ⤳ ofEquiv b ↔ b ≤ a

@[simp]

theorem Specialization.isOpen_toEquiv_preimage {α : Type u_1} [TopologicalSpace α] [AlexandrovDiscrete α] {s : Set (Specialization α)} : IsOpen (⇑toEquiv ⁻¹' s) ↔ IsUpperSet s

@[simp]

theorem Specialization.isUpperSet_ofEquiv_preimage {α : Type u_1} [TopologicalSpace α] [AlexandrovDiscrete α] {s : Set α} : IsUpperSet (⇑ofEquiv ⁻¹' s) ↔ IsOpen s

def Specialization.map {α : Type u_1} {β : Type u_2} [TopologicalSpace α] [TopologicalSpace β] (f : C(α, β)) : Specialization α →o Specialization β

A continuous map between topological spaces induces a monotone map between their specialization orders.

Equations

• Specialization.map f = { toFun := ⇑Specialization.toEquiv ∘ ⇑f ∘ ⇑Specialization.ofEquiv, monotone' := ⋯ }

@[simp]

theorem Specialization.map_id {α : Type u_1} [TopologicalSpace α] : map (ContinuousMap.id α) = OrderHom.id

@[simp]

theorem Specialization.map_comp {α : Type u_1} {β : Type u_2} {γ : Type u_3} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ] (g : C(β, γ)) (f : C(α, β)) : map (g.comp f) = (map g).comp (map f)

def orderIsoSpecializationWithUpperSetTopology (α : Type u_1) [Preorder α] : α ≃o Specialization (Topology.WithUpperSet α)

A preorder is isomorphic to the specialisation order of its upper set topology.

Equations

• orderIsoSpecializationWithUpperSetTopology α = { toEquiv := Topology.WithUpperSet.toUpperSet.trans Specialization.toEquiv, map_rel_iff' := ⋯ }

def homeoWithUpperSetTopologyorderIso (α : Type u_1) [TopologicalSpace α] [AlexandrovDiscrete α] : α ≃ₜ Topology.WithUpperSet (Specialization α)

An Alexandrov-discrete space is isomorphic to the upper set topology of its specialisation order.

Equations

• homeoWithUpperSetTopologyorderIso α = (Specialization.toEquiv.trans Topology.WithUpperSet.toUpperSet).toHomeomorph ⋯

def topToPreord : CategoryTheory.Functor TopCat Preord

Sends a topological space to its specialisation order.

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem topToPreord_obj_coe (X : TopCat) : ↑(topToPreord.obj X) = Specialization ↑X

@[simp]

theorem topToPreord_map {X✝ Y✝ : TopCat} (f : X✝ ⟶ Y✝) : topToPreord.map f = Preord.ofHom (Specialization.map (TopCat.Hom.hom f))