Duality principle of W. Sierpiński : in the abstract Baire-cathegory theory

Abstract

Let ^ be an 'JJJ-family of subsets of X and —the family of its “first category” sets. It is proven that one and only one of the following conditions is satisfied: (*) each ^i-set is at most countable; (**) X is the union o f^ i-se t and a set having property (L), which are disjoint; (***) each ^-residual set contains an uncountable ^fi-set. Moreover, a n d@ C 2 y are two 9J}-families, the “ duality principle” holds (i.e. there exists a bijection / : X -* Y transforming 'if,-sets onto ££>i-sets) ifF*^ and satisfy the same of the conditions above. Also, some considerations are added, concerning the coincidence between the properties of the family and a a — ideal.