Towards a descriptive theory of cb 0-spaces

The paper tries to extend some results of the classical Descriptive Set Theory to as many countably based T 0-spaces (cb0-spaces) as possible.Along with extending some central facts about Borel,Luzin and Hausdorff hierarchies of sets we also consider the more general case of k-partitions.In particular, we investigate the difference hierarchy of k-partitions and the fine hierarchy closely related to the Wadge hierarchy.