arXiv subjects
Publications and source records attributed to Ryszard Frankiewicz.
In this paper we present a simple proof of Gowers Dichotomy which states that every infinite dimensional Banach Space has a subspace which either contains an unconditional basic sequence or is hereditarily indecomposable. Our approach is purely combinatorial and mainly based on work of Ellentuck, Galvin and Prikry in infinite Ramsey theory.
It is proved that $P(ω)/\triangle_d$, where $\triangle_d$ is the ideal of sets of asymptotic density zero, is universal in the sense of embeddings.
The main result of this paper is to show that, if $κ$ is the smallest real-valued measurable cardinal not greater than $ 2^{\aleph_0}$, then there exists a complete metric space of cardinality not greater than $ 2^κ$ admitting a Kuratowski partition.
In this note we give equivalence of the existence of K-partitions with the existence of the precipitous ideal which is essentially topological. This way we strengthen the main result of Frankiewczi and Kunen (1987).
We investigate the properties of ideals associated with Kuratowski partitions of non-complete Baire metric spaces. We show that such an ideal can be precipitous.
A combinatorial proof of a pigeonhole principle of Gowers is found along with its symmetric and approximate version, FIN$_k^\pm$ theorem. The proofs do not use of the concept of ultrafilter.
We show that large sets in Ellentuck topology (i.e. sets which are not nowhere Ramsey) do not admit Kuratowski's partition. The similar result is true for the Sacks real forcing.
In this paper there is proved without any metamathematical techniques that the existence of precipitous ideals immediately follows from Kuratowski partitions.
In 1935 K. Kuratowski posed the problem whether a function f:X ! Y , (X is completely metrizable and Y is metrizable), with the property that a preimage of each open has the Baire property, is continuous apart from a meager set. This paper is a selection of older results related to the question posed by Kuratowski in 1935, coming from among others Solovay and Bukovsk?y, and quite new ones concerning considerations in Ellentuck topology and some propertiesof K-ideals, (i.e. ideals associated with Kuratowski partitions).
It is proved that no non-meager subspace of the space $[ω]^ω$ equipped with the Ellentuck topology does admit a Kuratowski partition, that is such a subset cannot be covered by a family $\mathfrak{F}$ of disjoint relatively meager sets such that $\bigcup\mathfrak{F}'$ has the Baire property (relatively) for every subfamily $\mathfrak{F}'\subseteq\mathfrak{F}$. It is also shown that the existence of Kuratowski partitions is not a metric problem. Some remarks concerning continuous restrictions of functions with domain in the Ellentuck space are made.
The structure of quotient Boolean algebras in terms of cardinal invariants is investigated. Some results of Gitik and Shelah regarding atomless ideals are reproved and proofs are significantly simplified.
It is proved that -- consistently -- there can be no ccc closed P-sets in the remainder space omega^* .