Theorems · Theorem · general topology
upperClosure_eq_Ici_csInf
∀ {α : Type u_3} [inst : ConditionallyCompleteLinearOrder α] [inst_1 : TopologicalSpace α] [OrderTopology α]
{s : Set α}, s.Nonempty → BddBelow s → IsClosed s → ↑(upperClosure s) = Set.Ici (sInf s)- Defined in
- Mathlib.Topology.Order.IsLUB
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites16
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
- SetLike.coestatement and proof · cited by 8,199
- Set.Nonemptystatement and proof · cited by 2,627
- Set.extproof · cited by 2,266
- IsClosedstatement and proof · cited by 1,639
- OrderTopologystatement and proof · cited by 1,355
- Set.Icistatement and proof · cited by 1,070
- InfSet.sInfstatement and proof · cited by 935
- ConditionallyCompleteLinearOrderstatement and proof · cited by 542
- BddBelowstatement and proof · cited by 401
- UpperSetstatement · cited by 245
Cited by2
Results whose statement or proof uses this declaration.
- IsClosed.upperClosureproof · cited by 1
- lowerClosure_eq_Iic_csSupproof · cited by 0