BW property for van der Waerden and Hindman ideals

We consider four classes of topological spaces defined with the aid of
convergence with respect to ideals on the set of natural numbers and
examine the relationships between each other.
We also show that two of those classes coincide for two particular ideals
connected with well known theorems in combinatorics, namely van der
Waerden's theorem and Hindman's theorem, and try to generalize the
results.