arXiv subjects
Publications and source records attributed to Sławomir Kusiński.
We first present a reduction of properties of compact sets of real numbers to properties of countable orders. Then discuss a variant of homogeneity of compact subsets of real numbers, focusing on the family of $t$-sets. Finally, we prove that there are exactly $ω_1$ many non-homeomorphic $t$-sets.
In this paper, we introduce product-wise generalizations of certain Marczewski-Burstin bases, including sets with the (s)-property and completely Ramsey sets. For each of these families, we establish analogs of the classical Luzin and Eggleston theorems, showing that functions measurable with respect to these families can be reduced to continuous functions on products of perfect sets. Furthermore, we provide a method for reducilng sequences of such functions to continuity, which allows us to generalize Laver's extension of Halpern-Läuchli and Harrington theorems.
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.
In this paper we provide solutions of the Harrington problem (along with a few generalizations) proposed in a book Analytic Sets. The original problem asks if for arbitrary sequence of continuous functions from \( \R^ω\) to a fixed compact interval we can find a subsequence point-wise convergent on some product of perfect subsets of \( \R \). We reduce aforementioned problem to functions from \( C^ω\) to \(C\), where \(C\) is a standard Cantor set as well as also provide solution to the problem with Baire functions in place of continuous ones. Our main focus is on showing applications of the fusion lemma - a result about perfect trees used among others to prove minimality of Sack's forcing - to the problem at hand.
In this paper we present a few properties of $K$-partitions, which are partitions of Baire spaces such that all subfamilies of such a partition sum to a set with the Baire property. Among the result proven we have general existence result that state that the existence of any $K$-partition implies the existence of $K$-partition of a metrizable space as well as existence of $K$-partition of a compact space implies the existence of $K$-partition of a completely metrizable space. We also prove some connections between existence of $K$-partitions and existence of precipitous ideals as well as measurable cardinals. There are also outlined possible connection with real-measurable cardinals, extensions of Lebesgue measure on the closed interval and density topologies.