Theorems · Theorem · dynamical systems

MulAction.isTopologicallyTransitive_iff_dense_iUnion_preimage

∀ (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 → Dense (⋃ m, (fun x => m • x) ⁻¹' U)

An action of a monoid M on α is topologically transitive if and only if for any nonempty open subset U of α the union of the preimages of U over the elements of M is dense in α.

Defined in

Mathlib.Dynamics.Transitive

Cited by

2 results in Mathlib

Foundations

Depth 65 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

TopologicalSpaceMonoidMulAction