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Theorems · Theorem · general topology

IsLowerSet.isClosed

∀ {α : Type u} [inst : TopologicalSpace α] [inst_1 : LinearOrder α] [OrderTopology α] [WellFoundedGT α] {s : Set α},
  IsLowerSet s → IsClosed s
Defined in
Mathlib.Topology.Order.Basic
Cited by
1 results in Mathlib
Foundations
Depth 75 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes
TopologicalSpaceLinearOrderOrderTopologyWellFoundedGT
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Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

IsUpperSet.isOpen · cited by 0IsUpperSet.isOpenSet · cited by 53352SetTopologicalSpace · cited by 24529TopologicalSpaceLinearOrder · cited by 8572LinearOrderIsClosed · cited by 1639IsClosedOrderTopology · cited by 1355OrderTopologySet.Iic · cited by 1111Set.IicIsLowerSet · cited by 167IsLowerSetWellFoundedGT · cited by 114WellFoundedGTisClosed_empty · cited by 26isClosed_emptyisClosed_Iic · cited by 23isClosed_IicIsLowerSet.eq_empty_or_Iic · cited by 2IsLowerSet.eq_empty_or_IicIsLowerSet.isClosedCITED BYCITES

Cites11

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Cited by1

Results whose statement or proof uses this declaration.