Theorems · Theorem · general topology

Topology.IsUpper.isTopologicalSpace_basis

∀ {α : Type u_1} [inst : CompleteLinearOrder α] [t : TopologicalSpace α] [Topology.IsUpper α] (U : Set α),
  IsOpen U ↔ U = Set.univ ∨ ∃ a, (Set.Iic a)ᶜ = U

Defined in: Mathlib.Topology.Order.LowerUpperTopology
Cited by: 1 results in Mathlib
Foundations: Depth 74 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CompleteLinearOrder TopologicalSpace Topology.IsUpper

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Cites9

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.univstatement · cited by 3,945
Compl.complstatement · cited by 2,925
IsOpenstatement · cited by 2,400
Set.Iicstatement · cited by 1,111
CompleteLinearOrderstatement and proof · cited by 126
Topology.IsUpperstatement and proof · cited by 18
Topology.IsLower.isTopologicalSpace_basisproof · cited by 1

Cited by1

Results whose statement or proof uses this declaration.

Topology.IsScott.scott_eq_upper_of_completeLinearOrderproof · cited by 0