Let C be an M-family of subsets of X and C1 - the family of its “first category” sets. It is proven that one and only one of the following conditions is satisfied: (*) each C1-set is at most countable; (**) X is the union of C1 set and a set having property (L), which are disjoint; (***) each C-residual set contains an uncountable C1-set.
Moreover, if C⊂2X and D⊂2Y are two M-families, the “duality principle” holds (i.e. there exists a bijection f: X→Y transforming C1-sets onto D1-sets) iff C and D satisfy the same of the conditions above.
Also, some considerations are added, concerning the coincidence between the properties of the family C1 and a σ-ideal.
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Vol. 2 (1986)
Published: 1986-09-30