Theorems · Theorem · general topology
Dense.nonempty
∀ {X : Type u} [inst : TopologicalSpace X] {s : Set X} [h : Nonempty X], Dense s → s.Nonempty- Defined in
- Mathlib.Topology.Closure
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 66 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- TopologicalSpaceNonempty
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites5
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 · cited by 2,627
- Densestatement and proof · cited by 359
- Dense.nonempty_iffproof · cited by 2
Cited by4
Results whose statement or proof uses this declaration.
- nonempty_interior_of_iUnion_of_closedproof · cited by 3
- Dense.ciSupproof · cited by 2
- Set.Countable.isPathConnected_compl_of_one_lt_rankproof · cited by 2
- not_isMeagre_of_isGδ_of_denseproof · cited by 1