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Mathlib.Data.Set.Intervals.OrdConnectedComponent

Order connected components of a set #

In this file we define Set.ordConnectedComponent s x to be the set of y such that Set.uIcc x y ⊆ s and prove some basic facts about this definition. At the moment of writing, this construction is used only to prove that any linear order with order topology is a T₅ space, so we only add API needed for this lemma.

def Set.ordConnectedComponent {α : Type u_1} [LinearOrder α] (s : Set α) (x : α) :
Set α

Order-connected component of a point x in a set s. It is defined as the set of y such that Set.uIcc x y ⊆ s. Note that it is empty if and only if x ∉ s.

Equations
Instances For
    theorem Set.mem_ordConnectedComponent {α : Type u_1} [LinearOrder α] {s : Set α} {x : α} {y : α} :
    theorem Set.dual_ordConnectedComponent {α : Type u_1} [LinearOrder α] {s : Set α} {x : α} :
    Set.ordConnectedComponent (OrderDual.ofDual ⁻¹' s) (OrderDual.toDual x) = OrderDual.ofDual ⁻¹' Set.ordConnectedComponent s x
    theorem Set.subset_ordConnectedComponent {α : Type u_1} [LinearOrder α] {s : Set α} {x : α} {t : Set α} [h : Set.OrdConnected s] (hs : x s) (ht : s t) :
    @[simp]
    theorem Set.self_mem_ordConnectedComponent {α : Type u_1} [LinearOrder α] {s : Set α} {x : α} :
    @[simp]
    @[simp]
    @[simp]
    theorem Set.ordConnectedComponent_univ {α : Type u_1} [LinearOrder α] {x : α} :
    Set.ordConnectedComponent Set.univ x = Set.univ
    theorem Set.mem_ordConnectedComponent_trans {α : Type u_1} [LinearOrder α] {s : Set α} {x : α} {y : α} {z : α} (hxy : y Set.ordConnectedComponent s x) (hyz : z Set.ordConnectedComponent s y) :
    theorem Set.ordConnectedComponent_eq {α : Type u_1} [LinearOrder α] {s : Set α} {x : α} {y : α} (h : Set.uIcc x y s) :
    Equations
    • One or more equations did not get rendered due to their size.
    noncomputable def Set.ordConnectedProj {α : Type u_1} [LinearOrder α] (s : Set α) :
    sα

    Projection from s : Set α to α sending each order connected component of s to a single point of this component.

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      @[simp]
      theorem Set.ordConnectedProj_eq {α : Type u_1} [LinearOrder α] {s : Set α} {x : s} {y : s} :
      def Set.ordConnectedSection {α : Type u_1} [LinearOrder α] (s : Set α) :
      Set α

      A set that intersects each order connected component of a set by a single point. Defined as the range of Set.ordConnectedProj s.

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        theorem Set.dual_ordConnectedSection {α : Type u_1} [LinearOrder α] (s : Set α) :
        Set.ordConnectedSection (OrderDual.ofDual ⁻¹' s) = OrderDual.ofDual ⁻¹' Set.ordConnectedSection s
        theorem Set.eq_of_mem_ordConnectedSection_of_uIcc_subset {α : Type u_1} [LinearOrder α] {s : Set α} {x : α} {y : α} (hx : x Set.ordConnectedSection s) (hy : y Set.ordConnectedSection s) (h : Set.uIcc x y s) :
        x = y
        def Set.ordSeparatingSet {α : Type u_1} [LinearOrder α] (s : Set α) (t : Set α) :
        Set α

        Given two sets s t : Set α, the set Set.orderSeparatingSet s t is the set of points that belong both to some Set.ordConnectedComponent tᶜ x, x ∈ s, and to some Set.ordConnectedComponent sᶜ x, x ∈ t. In the case of two disjoint closed sets, this is the union of all open intervals $(a, b)$ such that their endpoints belong to different sets.

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          theorem Set.dual_ordSeparatingSet {α : Type u_1} [LinearOrder α] {s : Set α} {t : Set α} :
          Set.ordSeparatingSet (OrderDual.ofDual ⁻¹' s) (OrderDual.ofDual ⁻¹' t) = OrderDual.ofDual ⁻¹' Set.ordSeparatingSet s t
          def Set.ordT5Nhd {α : Type u_1} [LinearOrder α] (s : Set α) (t : Set α) :
          Set α

          An auxiliary neighborhood that will be used in the proof of OrderTopology.t5Space.

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            theorem Set.disjoint_ordT5Nhd {α : Type u_1} [LinearOrder α] {s : Set α} {t : Set α} :