Topological facts about upper/lower/order-connected sets #
The topological closure and interior of an upper/lower/order-connected set is an upper/lower/order-connected set (with the notable exception of the closure of an order-connected set).
Implementation notes #
The same lemmas are true in the additive/multiplicative worlds. To avoid code duplication, we
provide HasUpperLowerClosure, an ad hoc axiomatisation of the properties we need.
Ad hoc class stating that the closure of an upper set is an upper set. This is used to state lemmas that do not mention algebraic operations for both the additive and multiplicative versions simultaneously. If you find a satisfying replacement for this typeclass, please remove it!
- isUpperSet_closure : ∀ (s : Set α), IsUpperSet s → IsUpperSet (closure s)
- isLowerSet_closure : ∀ (s : Set α), IsLowerSet s → IsLowerSet (closure s)
- isOpen_upperClosure : ∀ (s : Set α), IsOpen s → IsOpen ↑(upperClosure s)
- isOpen_lowerClosure : ∀ (s : Set α), IsOpen s → IsOpen ↑(lowerClosure s)
Instances
theorem
HasUpperLowerClosure.isUpperSet_closure
{α : Type u_1}
[TopologicalSpace α]
[Preorder α]
[self : HasUpperLowerClosure α]
(s : Set α)
:
IsUpperSet s → IsUpperSet (closure s)
theorem
HasUpperLowerClosure.isLowerSet_closure
{α : Type u_1}
[TopologicalSpace α]
[Preorder α]
[self : HasUpperLowerClosure α]
(s : Set α)
:
IsLowerSet s → IsLowerSet (closure s)
theorem
HasUpperLowerClosure.isOpen_upperClosure
{α : Type u_1}
[TopologicalSpace α]
[Preorder α]
[self : HasUpperLowerClosure α]
(s : Set α)
:
IsOpen s → IsOpen ↑(upperClosure s)
theorem
HasUpperLowerClosure.isOpen_lowerClosure
{α : Type u_1}
[TopologicalSpace α]
[Preorder α]
[self : HasUpperLowerClosure α]
(s : Set α)
:
IsOpen s → IsOpen ↑(lowerClosure s)
@[instance 100]
instance
OrderedAddCommGroup.to_hasUpperLowerClosure
{α : Type u_1}
[TopologicalSpace α]
[OrderedAddCommGroup α]
[ContinuousConstVAdd α α]
:
Equations
- ⋯ = ⋯
@[instance 100]
instance
OrderedCommGroup.to_hasUpperLowerClosure
{α : Type u_1}
[TopologicalSpace α]
[OrderedCommGroup α]
[ContinuousConstSMul α α]
:
Equations
- ⋯ = ⋯
theorem
IsUpperSet.closure
{α : Type u_1}
[TopologicalSpace α]
[Preorder α]
[HasUpperLowerClosure α]
{s : Set α}
:
IsUpperSet s → IsUpperSet (closure s)
theorem
IsLowerSet.closure
{α : Type u_1}
[TopologicalSpace α]
[Preorder α]
[HasUpperLowerClosure α]
{s : Set α}
:
IsLowerSet s → IsLowerSet (closure s)
theorem
IsOpen.upperClosure
{α : Type u_1}
[TopologicalSpace α]
[Preorder α]
[HasUpperLowerClosure α]
{s : Set α}
:
IsOpen s → IsOpen ↑(upperClosure s)
theorem
IsOpen.lowerClosure
{α : Type u_1}
[TopologicalSpace α]
[Preorder α]
[HasUpperLowerClosure α]
{s : Set α}
:
IsOpen s → IsOpen ↑(lowerClosure s)
instance
instHasUpperLowerClosureOrderDual
{α : Type u_1}
[TopologicalSpace α]
[Preorder α]
[HasUpperLowerClosure α]
:
Equations
- ⋯ = ⋯
theorem
IsUpperSet.interior
{α : Type u_1}
[TopologicalSpace α]
[Preorder α]
[HasUpperLowerClosure α]
{s : Set α}
(h : IsUpperSet s)
:
IsUpperSet (interior s)
theorem
IsLowerSet.interior
{α : Type u_1}
[TopologicalSpace α]
[Preorder α]
[HasUpperLowerClosure α]
{s : Set α}
(h : IsLowerSet s)
:
IsLowerSet (interior s)
theorem
Set.OrdConnected.interior
{α : Type u_1}
[TopologicalSpace α]
[Preorder α]
[HasUpperLowerClosure α]
{s : Set α}
(h : s.OrdConnected)
:
(interior s).OrdConnected