Theorems · Theorem · general topology
isConnected_uIoo
∀ {α : Type u} [inst : TopologicalSpace α] [inst_1 : ConditionallyCompleteLinearOrder α] [OrderTopology α]
[DenselyOrdered α] {a b : α}, a ≠ b → IsConnected (Set.uIoo a b)- Defined in
- Mathlib.Topology.Order.IntermediateValue
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 90 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites8
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- TopologicalSpacestatement and proof · cited by 24,529
- OrderTopologystatement and proof · cited by 1,355
- ConditionallyCompleteLinearOrderstatement and proof · cited by 542
- DenselyOrderedstatement and proof · cited by 471
- IsConnectedstatement · cited by 116
- Set.uIoostatement · cited by 68
- isPreconnected_uIooproof · cited by 1
- Set.nonempty_uIooproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- exists_eq_interval_average_of_nullSingletonClassproof · cited by 2