Order-connected sets #
We say that a set s : Set α
is OrdConnected
if for all x y ∈ s
it includes the
interval [[x, y]]
. If α
is a DenselyOrdered
ConditionallyCompleteLinearOrder
with
the OrderTopology
, then this condition is equivalent to IsPreconnected s
. If α
is a
LinearOrderedField
, then this condition is also equivalent to Convex α s
.
In this file we prove that intersection of a family of OrdConnected
sets is OrdConnected
and
that all standard intervals are OrdConnected
.
theorem
Set.ordConnected_of_Ioo
{α : Type u_3}
[PartialOrder α]
{s : Set α}
(hs : ∀ x ∈ s, ∀ y ∈ s, x < y → Set.Ioo x y ⊆ s)
:
s.OrdConnected
theorem
Set.ordConnected_Icc
{α : Type u_1}
[Preorder α]
{a : α}
{b : α}
:
(Set.Icc a b).OrdConnected
theorem
Set.ordConnected_Ico
{α : Type u_1}
[Preorder α]
{a : α}
{b : α}
:
(Set.Ico a b).OrdConnected
theorem
Set.ordConnected_Ioc
{α : Type u_1}
[Preorder α]
{a : α}
{b : α}
:
(Set.Ioc a b).OrdConnected
theorem
Set.ordConnected_Ioo
{α : Type u_1}
[Preorder α]
{a : α}
{b : α}
:
(Set.Ioo a b).OrdConnected
instance
Set.instDenselyOrdered
{α : Type u_1}
[Preorder α]
[DenselyOrdered α]
{s : Set α}
[hs : s.OrdConnected]
:
In a dense order α
, the subtype from an OrdConnected
set is also densely ordered.
Equations
- ⋯ = ⋯
theorem
Set.ordConnected_range
{α : Type u_1}
{β : Type u_2}
[Preorder α]
[Preorder β]
{E : Type u_3}
[EquivLike E α β]
[OrderIsoClass E α β]
(e : E)
:
(Set.range ⇑e).OrdConnected
theorem
IsAntichain.ordConnected
{α : Type u_1}
[PartialOrder α]
{s : Set α}
(hs : IsAntichain (fun (x1 x2 : α) => x1 ≤ x2) s)
:
s.OrdConnected
theorem
Set.ordConnected_inter_Icc_of_subset
{α : Type u_1}
[PartialOrder α]
{s : Set α}
{x : α}
{y : α}
(h : Set.Ioo x y ⊆ s)
:
theorem
Set.ordConnected_uIcc
{α : Type u_1}
[LinearOrder α]
{a : α}
{b : α}
:
(Set.uIcc a b).OrdConnected
theorem
Set.OrdConnected.uIcc_subset
{α : Type u_1}
[LinearOrder α]
{s : Set α}
(hs : s.OrdConnected)
⦃x : α⦄
(hx : x ∈ s)
⦃y : α⦄
(hy : y ∈ s)
:
theorem
Set.OrdConnected.uIoc_subset
{α : Type u_1}
[LinearOrder α]
{s : Set α}
(hs : s.OrdConnected)
⦃x : α⦄
(hx : x ∈ s)
⦃y : α⦄
(hy : y ∈ s)
:
theorem
Set.ordConnected_of_uIcc_subset_left
{α : Type u_1}
[LinearOrder α]
{s : Set α}
{x : α}
(h : ∀ y ∈ s, Set.uIcc x y ⊆ s)
:
s.OrdConnected
theorem
Set.ordConnected_iff_uIcc_subset_left
{α : Type u_1}
[LinearOrder α]
{s : Set α}
{x : α}
(hx : x ∈ s)
:
theorem
Set.ordConnected_iff_uIcc_subset_right
{α : Type u_1}
[LinearOrder α]
{s : Set α}
{x : α}
(hx : x ∈ s)
: