Theorems · Theorem · general topology
Set.PairwiseDisjoint.countable_of_nonempty_interior
∀ {α : Type u} [t : TopologicalSpace α] [TopologicalSpace.SeparableSpace α] {ι : Type u_2} {s : ι → Set α} {a : Set ι},
a.PairwiseDisjoint s → (∀ i ∈ a, (interior (s i)).Nonempty) → a.CountableIn a separable space, a family of disjoint sets with nonempty interiors is countable.
- Defined in
- Mathlib.Topology.Bases
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 68 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
- Set.Nonemptystatement and proof · cited by 2,627
- interiorstatement and proof · cited by 714
- Set.Countablestatement · cited by 545
- Set.PairwiseDisjointstatement and proof · cited by 275
- interior_subsetproof · cited by 171
- isOpen_interiorproof · cited by 130
- TopologicalSpace.SeparableSpacestatement and proof · cited by 109
- Set.PairwiseDisjoint.monoproof · cited by 8
- Set.PairwiseDisjoint.countable_of_isOpenproof · cited by 4
Cited by4
Results whose statement or proof uses this declaration.
- VitaliFamily.FineSubfamilyOn.index_countableproof · cited by 2
- Vitali.exists_disjoint_covering_aeproof · cited by 1
- Besicovitch.exist_finset_disjoint_balls_large_measureproof · cited by 1
- Besicovitch.exists_closedBall_covering_tsum_measure_leproof · cited by 1