Theorems · Theorem · general topology
IsClosed.csSup_mem
∀ {α : Type u_1} [inst : ConditionallyCompleteLinearOrder α] [inst_1 : TopologicalSpace α] [OrderTopology α]
{s : Set α}, IsClosed s → s.Nonempty → BddAbove s → sSup s ∈ s- Defined in
- Mathlib.Topology.Order.Monotone
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 82 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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.Nonemptystatement and proof · cited by 2,627
- IsClosedstatement and proof · cited by 1,639
- OrderTopologystatement and proof · cited by 1,355
- SupSet.sSupstatement · cited by 954
- BddAbovestatement and proof · cited by 620
- ConditionallyCompleteLinearOrderstatement and proof · cited by 542
- isLUB_csSupproof · cited by 34
- IsLUB.mem_of_isClosedproof · cited by 3
Cited by2
Results whose statement or proof uses this declaration.
- IsClosed.mem_of_ge_of_forall_exists_gtproof · cited by 2
- eq_Icc_csInf_csSup_of_connected_bdd_closedproof · cited by 1