Theorems · Theorem · general topology
lowerClosure_eq_Iic_csSup
∀ {α : Type u_3} [inst : ConditionallyCompleteLinearOrder α] [inst_1 : TopologicalSpace α] [OrderTopology α]
{s : Set α}, s.Nonempty → BddAbove s → IsClosed s → ↑(lowerClosure s) = Set.Iic (sSup s)- Defined in
- Mathlib.Topology.Order.IsLUB
- Cited by
- 0 results in Mathlib
- Foundations
- Depth 84 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites13
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 · cited by 8,199
- Set.Nonemptystatement and proof · cited by 2,627
- IsClosedstatement and proof · cited by 1,639
- OrderTopologystatement and proof · cited by 1,355
- Set.Iicstatement · cited by 1,111
- SupSet.sSupstatement · cited by 954
- BddAbovestatement and proof · cited by 620
- ConditionallyCompleteLinearOrderstatement and proof · cited by 542
- LowerSetstatement · cited by 230
- lowerClosurestatement · cited by 83
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