Let ???? be an ????-family of subsets of X and ????1 - the family of its “first category” sets. It is proven that one and only one of the following conditions is satisfied: (*) each ????1-set is at most countable; (**) X is the union of ????1 set and a set having property (L), which are disjoint; (***) each ????-residual set contains an uncountable ????1-set.
Moreover, if ????⊂2X and ????⊂2Y are two ????-families, the “duality principle” holds (i.e. there exists a bijection f: X→Y transforming ????1-sets onto ????1-sets) iff ???? and ???? satisfy the same of the conditions above.
Also, some considerations are added, concerning the coincidence between the properties of the family ????1 and a σ-ideal.
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Vol. 2 (1986)
Published: 1986-09-30
10.1515/amsil
English