A note on hierarchies of Borel type sets

The paper deals with classes of subsets, that is classes consisting of pairs ( Q , X ) , where Q is a subset of a space X. The main result of the paper concerns the so-called hereditary complete saturated classes of subsets. For such a class it is proved that there exist a space X and a pair ( Q , X ) ∈ P for which ( X ∖ Q , X ) ∉ P . tary complete saturated classes of subsets are, for example, classes consisting of the pairs ( Q , X ) , where Q is a Borel type set of a space X of the additive class or of the multiplicative class α. Borel type sets are obtained from the open sets by the same process as the Borel sets of a metrizable space replacing the countable sums and countable intersections by sums and intersections of τ many members, where τ is an infinite cardinal. From the main result of the paper it follows the well-known result that in the Cantor cub D τ for every α ∈ τ + there exists a Borel type set of the additive class α which is not a Borel type set of the class β < α . oof of the main result consists of two steps. In the first step it is constructed a corresponding universal set and in the second it is used the Cantor diagonal theorem. The method of construction of the universal set is new even for the case τ = ω .