  Using the combinatorial properties of subsets of integers, a classification
of metric dynamical systems was given in [V. Bergelson and T. Downarowicz,
Large sets of integers and hierarchy of mixing properties of measure-preserving
systems, Colloquium Mathematicum, 110(1), (2008), 117-150]. As a result, some
new families emerged. Here, their counterparts in topological dynamics has been
considered. The differences will be discussed and new classes of systems in
topological dynamics will be introduced.
