A note on hierarchies of Borel type sets

• Published in: Topology and its applications 2012-04, Vol.159 (7), p.1702-1704, Article 1702
• Main Author: Iliadis, S.D.
• Format: Article
• Language: English
• Subjects: Borel type set; Hierarchies of Borel type sets; Regular space; Saturated class of subsets
• ISSN: 0166-8641; 1879-3207
• DOI: 10.1016/j.topol.2011.09.041

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. Hereditary 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 Dτ 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