Theorems · Theorem · order theory
SuccOrder.isOpen_singleton_iff
∀ {α : Type u_1} [inst : LinearOrder α] [inst_1 : TopologicalSpace α] [OrderTopology α] {a : α} [SuccOrder α]
[NoMaxOrder α], IsOpen {a} ↔ ¬Order.IsSuccLimit a- Defined in
- Mathlib.Topology.Order.SuccPred
- Cited by
- 3 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.
Cites29
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
- LinearOrderstatement and proof · cited by 8,572
- Nontrivialproof · cited by 2,416
- IsOpenstatement and proof · cited by 2,400
- OrderTopologystatement and proof · cited by 1,355
- Set.Iooproof · cited by 1,214
- Set.Iioproof · cited by 1,166
- LE.le.trans_ltproof · cited by 795
- Order.succproof · cited by 633
- SuccOrderstatement and proof · cited by 574
- IsOpen.mem_nhdsproof · cited by 470
Cited by3
Results whose statement or proof uses this declaration.
- SuccOrder.nhds_eq_pureproof · cited by 3
- SuccOrder.isSuccLimit_of_mem_frontierproof · cited by 1
- Ordinal.isOpen_singleton_iffproof · cited by 0