Theorems · Theorem · general topology
closure_inter_open_nonempty_iff
∀ {X : Type u} [inst : TopologicalSpace X] {s t : Set X}, IsOpen t → ((closure s ∩ t).Nonempty ↔ (s ∩ t).Nonempty)- Defined in
- Mathlib.Topology.Closure
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 63 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites10
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
- IsOpenstatement and proof · cited by 2,400
- closurestatement and proof · cited by 1,254
- subset_closureproof · cited by 309
- Set.inter_commproof · cited by 291
- Set.Nonempty.monoproof · cited by 88
- inf_le_inf_rightproof · cited by 23
- mem_closure_iffproof · cited by 15
Cited by2
Results whose statement or proof uses this declaration.
- isPreirreducible_iff_closureproof · cited by 2
- Topology.RelCWComplex.disjoint_interior_base_closedCellproof · cited by 1