Theorems · Theorem · dynamical systems
IsOpen.dense_iUnion_preimage_smul
∀ (M : Type u_1) {α : Type u_2} [inst : TopologicalSpace α] [inst_1 : Monoid M] [inst_2 : MulAction M α]
[h : MulAction.IsTopologicallyTransitive M α] {U : Set α}, IsOpen U → U.Nonempty → Dense (⋃ m, (fun x => m • x) ⁻¹' U)- Defined in
- Mathlib.Dynamics.Transitive
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 66 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.preimagestatement · cited by 4,946
- Monoidstatement and proof · cited by 3,887
- Set.Nonemptystatement and proof · cited by 2,627
- Set.iUnionstatement · cited by 2,483
- IsOpenstatement and proof · cited by 2,400
- MulActionstatement and proof · cited by 1,294
- Densestatement · cited by 359
- MulAction.IsTopologicallyTransitivestatement and proof · cited by 8
- MulAction.isTopologicallyTransitive_iff_dense_iUnion_preimageproof · cited by 2
Cited by1
Results whose statement or proof uses this declaration.
- IsOpen.dense_of_preimage_smul_invariantproof · cited by 1