Duality principle of W. Sierpiński in the abstract Baire-cathegory theory
Abstract
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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Instytut Matematyki, Uniwersytet Śląski w Katowicach Poland
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