Theorems · Theorem · general topology
eq_Icc_csInf_csSup_of_connected_bdd_closed
∀ {α : Type u} [inst : TopologicalSpace α] [inst_1 : ConditionallyCompleteLinearOrder α] [OrderTopology α] {s : Set α},
IsConnected s → BddBelow s → BddAbove s → IsClosed s → s = Set.Icc (sInf s) (sSup s)- Defined in
- Mathlib.Topology.Order.IntermediateValue
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- TopologicalSpacestatement and proof · cited by 24,529
- Set.Iccstatement · cited by 1,702
- IsClosedstatement and proof · cited by 1,639
- OrderTopologystatement and proof · cited by 1,355
- SupSet.sSupstatement · cited by 954
- InfSet.sInfstatement · cited by 935
- BddAbovestatement and proof · cited by 620
- ConditionallyCompleteLinearOrderstatement and proof · cited by 542
- LE.le.antisymmproof · cited by 507
- BddBelowstatement and proof · cited by 401
- IsConnectedstatement and proof · cited by 116
Cited by1
Results whose statement or proof uses this declaration.
- eq_Icc_of_connected_compactproof · cited by 1