Theorems · Theorem · order theory
Topology.IsUpperSet.closure_singleton
∀ {α : Type u_1} [inst : Preorder α] [inst_1 : TopologicalSpace α] [Topology.IsUpperSet α] {a : α},
closure {a} = Set.Iic aThe closure of a singleton {a} in the upper set topology is the right-closed left-infinite
interval $(-∞,a]$.
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 67 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
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.coeproof · cited by 8,199
- Preorderstatement and proof · cited by 7,952
- closurestatement · cited by 1,254
- Set.Iicstatement and proof · cited by 1,111
- LowerSetproof · cited by 230
- LowerSet.Iicproof · cited by 37
- Topology.IsUpperSetstatement and proof · cited by 32
- lowerClosure_singletonproof · cited by 8
- Topology.IsUpperSet.closure_eq_lowerClosureproof · cited by 3
Cited by3
Results whose statement or proof uses this declaration.
- Topology.WithUpperSet.toUpperSet_specializes_toUpperSetproof · cited by 1
- Topology.IsUpperSet.monotone_iff_continuousproof · cited by 1
- Topology.IsUpperSet.specializes_iff_leproof · cited by 0