Convergence in van der Waerden and Hindman spaces

We consider four classes of topological spaces which are defined with the aid of convergence with respect to ideals on NN. All these classes are subclasses of countably compact spaces, and two of them are also subclasses of sequentially compact spaces. In the first part of the paper (Sections 1 and 2) we prove some properties of these classes. In the second part of the paper (Sections 3 and 4) we focus on spaces defined by two particular ideals connected with well known theorems in combinatorics, namely van der Waerden's theorem and Hindman's theorem. The main aim of this part of the paper is to show that two classes of the considered spaces coincide for the van der Waerden ideal and Hindman ideal respectively.