Interaction of separation properties with connectedness properties #
@[instance 100]
instance
TotallyDisconnectedSpace.t1Space
{X : Type u_1}
[TopologicalSpace X]
[h : TotallyDisconnectedSpace X]
:
T1Space X
theorem
PreconnectedSpace.trivial_of_discrete
{X : Type u_1}
[TopologicalSpace X]
[PreconnectedSpace X]
[DiscreteTopology X]
:
theorem
IsPreconnected.infinite_of_nontrivial
{X : Type u_1}
[TopologicalSpace X]
[T1Space X]
{s : Set X}
(h : IsPreconnected s)
(hs : s.Nontrivial)
:
s.Infinite
theorem
PreconnectedSpace.infinite
{X : Type u_1}
[TopologicalSpace X]
[PreconnectedSpace X]
[Nontrivial X]
[T1Space X]
:
Infinite X
@[deprecated PreconnectedSpace.infinite (since := "2025-03-21")]
theorem
ConnectedSpace.infinite
{X : Type u_1}
[TopologicalSpace X]
[PreconnectedSpace X]
[Nontrivial X]
[T1Space X]
:
Infinite X
Alias of PreconnectedSpace.infinite
.
@[instance 100]
instance
ConnectedSpace.neBot_nhdsWithin_compl_of_nontrivial_of_t1space
{X : Type u_1}
[TopologicalSpace X]
[ConnectedSpace X]
[Nontrivial X]
[T1Space X]
(x : X)
:
(nhdsWithin x {x}ᶜ).NeBot
A non-trivial connected T1 space has no isolated points.