Theorems · Theorem · general topology
minimal_nonempty_closed_subsingleton
∀ {X : Type u_1} [inst : TopologicalSpace X] [T0Space X] {s : Set X},
IsClosed s → (∀ t ⊆ s, t.Nonempty → IsClosed t → t = s) → s.Subsingleton- Defined in
- Mathlib.Topology.Separation.Basic
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 81 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpaceT0Space
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
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
- Set.Nonemptystatement and proof · cited by 2,627
- IsOpenproof · cited by 2,400
- IsClosedstatement and proof · cited by 1,639
- Set.Subsingletonstatement · cited by 276
- T0Spacestatement and proof · cited by 179
- Set.sdiff_subsetproof · cited by 156
- Eq.subsetproof · cited by 124
- Xorproof · cited by 56
- of_not_notproof · cited by 51
- IsClosed.sdiffproof · cited by 7
Cited by1
Results whose statement or proof uses this declaration.
- minimal_nonempty_closed_eq_singletonproof · cited by 2