Theorems · Theorem · dynamical systems

IsOpen.dense_of_preimage_smul_invariant

∀ (M : Type u_1) {α : Type u_2} [inst : TopologicalSpace α] [inst_1 : Monoid M] [inst_2 : MulAction M α]
  [MulAction.IsTopologicallyTransitive M α] {U : Set α},
  IsOpen U → U.Nonempty → (∀ (m : M), (fun x => m • x) ⁻¹' U ⊆ U) → Dense U

Let M be a monoid with a topologically transitive action on α. If U is a nonempty open subset of α and (m • ·) ⁻¹' U ⊆ U for all m : M then U is dense in α.

Defined in

Mathlib.Dynamics.Transitive

Cited by

1 results in Mathlib

Foundations

Depth 68 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceMonoidMulActionMulAction.IsTopologicallyTransitive