Physical Sciences and Mathematics Commons^™

Algebras With Inner Mb-Representation, Krzysztof Ciesielski

Faculty & Staff Scholarship

We investigate algebras of sets, and pairs (A,I) consisting of an algebra A and an ideal I, which is a subset of A, that possess an inner MB-representation. We compare inner MB-representability of (A,I) with several properties of (A,I) considered by Baldwin. We show that A is inner MB-representable if and only if A =S(A \ H (A)), where S(^.) is a Marczewski operation defined below and H consists of sets that are hereditarily in A. We study uniqueness issue of the ideal in that representation.

Continuous Images Of Big Sets And Additivity Of S0 Under Cpaprism, Krzysztof Ciesielski

Faculty & Staff Scholarship

We prove that the Covering Property Axiom CPA_prism, which holds in the iterated perfect set model, implies the following facts.

• There exists a family G of uniformly continuous functions from R to [0,1] such that G has cardinality \omega₁ < \continuum and for every subset S of R of cardinality \continuum there exists a g in G with g[S]=[0,1].
• The additivity of the Marczewski's ideal s₀ is equal to \omega₁ < \continuum.