Theorems · Theorem · general topology
Dense.inter_of_isOpen_left
∀ {X : Type u} [inst : TopologicalSpace X] {s t : Set X}, Dense s → Dense t → IsOpen s → Dense (s ∩ t)The intersection of an open dense set with a dense set is a dense set.
- Defined in
- Mathlib.Topology.Neighborhoods
- Cited by
- 7 results in Mathlib
- Foundations
- Depth 66 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
- IsOpenstatement and proof · cited by 2,400
- closureproof · cited by 1,254
- Densestatement and proof · cited by 359
- Set.inter_univproof · cited by 198
- isClosed_closureproof · cited by 195
- closure_minimalproof · cited by 94
- Dense.closure_eqproof · cited by 24
- IsOpen.inter_closureproof · cited by 8
Cited by7
Results whose statement or proof uses this declaration.
- Dense.inter_of_isOpen_rightproof · cited by 4
- AlgebraicGeometry.Scheme.PartialMap.equiv_iff_of_isSeparated_of_leproof · cited by 2
- AlgebraicGeometry.Scheme.PartialMap.equiv.transproof · cited by 2
- AlgebraicGeometry.Scheme.PartialMap.equiv_iff_of_isSeparatedstatement and proof · cited by 1
- AlgebraicGeometry.Scheme.PartialMap.equiv_of_fromSpecStalkOfMem_eqproof · cited by 1
- IsGδ.baireSpace_of_denseproof · cited by 1