Theorems · Theorem · general topology
isPreconnected_Icc
∀ {α : Type u} [inst : TopologicalSpace α] [inst_1 : ConditionallyCompleteLinearOrder α] [OrderTopology α]
[DenselyOrdered α] {a b : α}, IsPreconnected (Set.Icc a b)A closed interval in a densely ordered conditionally complete linear order is preconnected.
- Defined in
- Mathlib.Topology.Order.IntermediateValue
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 85 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setproof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Set.Nonemptyproof · cited by 2,627
- Set.Iccstatement and proof · cited by 1,702
- IsClosedproof · cited by 1,639
- OrderTopologystatement and proof · cited by 1,355
- ConditionallyCompleteLinearOrderstatement and proof · cited by 542
- DenselyOrderedstatement and proof · cited by 471
- le_totalproof · cited by 294
- Set.inter_commproof · cited by 291
- IsPreconnectedstatement · cited by 205
- Set.union_commproof · cited by 99
Cited by7
Results whose statement or proof uses this declaration.
- intermediate_value_Iccproof · cited by 4
- isPreconnected_uIccproof · cited by 3
- ContinuousOn.image_Iccproof · cited by 3
- intermediate_value_Icc'proof · cited by 3
- isConnected_Iccproof · cited by 2
- exists_mem_Icc_isFixedPtproof · cited by 1
- exists_mem_Icc_isFixedPt_of_surjOnproof · cited by 1