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Piotr Borodulin-Nadzieja

Publications and source records attributed to Piotr Borodulin-Nadzieja.

At least 19 recordsLinked to original sources
arXiv2026

Non-separable combinatorial Banach spaces

We study combinatorial Banach spaces, i.e. Banach spaces induced by families of finite sets, of uncountable density. We prove some general theorems, for example we characterize when such spaces contain isomorphic copies of long $c_0$ and long $\ell_1$. We discuss the existence of Banach spaces which are (complementedly) universal in the category of Banach spaces with unconditional basis of fixed cardinality. We present some examples, in particular we show that the Banach spaces generated by Suslin trees do not contain uncountable equilateral sets.

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arXiv2026

Geometric duality, perfect graphs, and the Sierpiński space

In their classical paper \emph{On the stopping time Banach space}, Bang and Odell, among a plethora of results concerning the dyadic stopping time space and its dual, presented the first non-trivial example of the \emph{duality phenomenon} between combinatorial Banach spaces. We give a full characterization of such pairs $(\mc{F}_0, \mc{F}_1)$ of families of finite sets: This duality holds iff there is a perfect graph $G$ on $\NN$ such that $\mc{F}_0$ consists of all finite cliques of $G$ and $\mc{F}_1$ consists of all finite anti-cliques of $G$. As it turns out, Lovász' famous perfect graph theorem is an immediate corollary of this result. Among the many examples of such pairs of families, we investigate a particularly interesting one, when $G$ is the Sierpiński graph, and study general methods of embedding combinatorial and classical sequence spaces in the generated space, including the Schreier and $\ell_p$ spaces.

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arXiv2026

On the Rudin-Blass Ordering of Measures

We study the Rudin-Blass (and the Rudin-Keisler) ordering on the finite additive measures on $ω$. We propose a generalization of the notion of Q-point and selective ultrafilter to measures: Q-measures and selective measures. We show some symmetries between Q-points and Q-measures but also we show where those symmetries break up. In particular we present an example of a measure which is minimal in the sense of Rudin-Blass but which is not a Q-measure.

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arXiv2025

On the spaces dual to combinatorial Banach spaces

We present quasi-Banach spaces which are closely related to the duals of combinatorial Banach spaces. More precisely, for a compact family $\mathcal{F}$ of finite subsets of $ω$ we define a quasi-norm $\lVert \cdot \rVert^\mathcal{F}$ whose Banach envelope is the dual norm for the combinatorial space generated by $\mathcal{F}$. Such quasi-norms seem to be much easier to handle than the dual norms and yet the quasi-Banach spaces induced by them share many properties with the dual spaces. We show that the quasi-Banach spaces induced by large families (in the sense of Lopez-Abad and Todorcevic) are $\ell_1$-saturated and do not have the Schur property. In particular, this holds for the Schreier families.

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arXiv2024

The ZOO of combinatorial Banach spaces

We study Banach spaces induced by families of finite sets in the most natural (Schreier-like) way, that is, we consider the completion $X_\mc{F}$ of $c_{00}$ with respect to the norm $\sup\{\sum_{k\in F}|x(k)|:F\in\mc{F}\}$ where $\mc{F}$ is an arbitrary (not necessarily compact) family of finite sets covering $\mbb{N}$. Among other results, we discuss the following: (1) Structure theorems bonding the combinatorics of $\mc{F}$ and the geometry of $X_\mc{F}$ including possible characterizations and variants of the Schur property, $\ell_1$-saturation, and the lack of copies of $c_0$ in $X_\mc{F}$. (2) A plethora of examples including a relatively simple $\ell_1$-saturated combinatorial space which does not satisfy the Schur property, as well as a new presentation of Pełczyński's universal space. (3) The complexity of the family $\{H\subseteq\NN:X_{\mc{F}\upharpoonright H}$ does not contain $c_0\}$.

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arXiv2024

P-measures in models without P-points

We answer in negative the problem if the existence of a P-measure implies the existence of a P-point. Namely, we show that if we add random reals to a certain unique P-point model, then in the resulting model we will have a P-measure but not P-points. Also, we investigate the question if there is a P-measure in the Silver model. We show that rapid filters cannot be extended to a P-measure in the extension by $ω$ product of Silver forcings and that in the model obtained by the product of $ω_2$ many Silver forcings there are no P-measures of countable Maharam type

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arXiv2023

On points avoiding measures

We say that an element $x$ of a topological space $X$ avoids measures if for every Borel measure $μ$ on $X$ if $μ(\{x\})=0$, then there is an open $U\ni x$ such that $μ(U)=0$. The negation of this property can viewed as a local version of the property of supporting a strictly positive measure. We study points avoiding measures in the general setting as well as in the context of $ω^\ast$, the remainder of Stone-Čech compactification of $ω$.

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arXiv2022

There is a P-measure in the random model

We say that a finitely additive probability measure $μ$ on $ω$ is \emph{a P-measure} if it vanishes on points and for each decreasing sequence $(E_n)$ of infinite subsets of $ω$ there is $E\subseteqω$ such that $E\subseteq^* E_n$ for each $n\inω$ and $μ(E) = \lim_{n\to\infty}μ(E_n)$. Thus, P-measures generalize in a natural way P-points and it is known that, similarly as in the case of P-points, their existence is independent of $\mathsf{ZFC}$. In this paper we show that there is a P-measure in the model obtained by adding any number of random reals to a model of $\mathsf{CH}$. As a corollary, we obtain that in the classical random model $ω^*$ contains a nowhere dense ccc closed P-set.

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arXiv2021

On measures induced by forcing names for ultrafilters

We study the interplay between properties of measures on a Boolean algebra A and forcing names for ultrafilters on A. We show that several well known measure theoretic properties of Boolean algebras (such as supporting a strictly positive measure or carrying only separable measures) have quite natural characterizations in the forcing language. We show some applications of this approach. In particular, we reprove a theorem of Kunen saying that in the classical random model there are no towers of height $ω_2$.

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arXiv2021

On sequences of homomorphisms into measure algebras and the Efimov Problem

For given Boolean algebras $\mathbb{A}$ and $\mathbb{B}$ we endow the space $\mathcal{H}(\mathbb{A},\mathbb{B})$ of all Boolean homomorphisms from $\mathbb{A}$ to $\mathbb{B}$ with various topologies and study convergence properties of sequences in $\mathcal{H}(\mathbb{A},\mathbb{B})$. We are in particular interested in the situation when $\mathbb{B}$ is a measure algebra as in this case we obtain a natural tool for studying topological convergence properties of sequences of ultrafilters on $\mathbb{A}$ in random extensions of the set-theoretical universe. This appears to have strong connections with Dow and Fremlin's result stating that there are Efimov spaces in the random model. We also investigate relations between topologies on $\mathcal{H}(\mathbb{A},\mathbb{B})$ for a Boolean algebra $\mathbb{B}$ carrying a strictly positive measure and convergence properties of sequences of measures on $\mathbb{A}$.

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arXiv2019

Splitting chains, tunnels and twisted sums

We study splitting chains in $\mathscr{P}(ω)$, that is, families of subsets of $ω$ which are linearly ordered by $\subseteq^*$ and which are splitting. We prove that their existence is independent of axioms of $\mathsf{ZFC}$. We show that they can be used to construct certain peculiar Banach spaces: twisted sums of $C(ω^*)=\ell_\infty/c_0$ and $c_0(\mathfrak c)$. Also, we consider splitting chains in a topological setting, where they give rise to the so called tunnels.

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arXiv2019

Analytic P-ideals and Banach spaces

We study the interplay between Banach space theory and theory of analytic P-ideals. Applying the observation that, up to isomorphism, all Banach spaces with unconditional bases can be constructed in a way very similar to the construction of analytic P-ideals from submeasures, we point out numerous symmetries between the two theories. Also, we investigate a special case, the interactions between combinatorics of families of finite sets, topological properties of the "Schreier type" Banach spaces associated to these families, and the complexity of ideals generated by the canonical bases in these spaces. Among other results, we present some new examples of Banach spaces and analytic P-ideals, a new characterization of precompact families and its applications to enhance Pták's Lemma and Mazur's Lemma.

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arXiv2016

Measures and slaloms

We examine measure-theoretic properties of spaces constructed using certain technique of Todorčević. We show that the existence of strictly positive measures on such spaces depends on combinatorial properties of certain families of slaloms. As a corollary we get that if $\mathrm{add}(\mathcal{N}) = \mathrm{non}(\mathcal{M})$ then there is a non-separable space which supports a measure and which cannot be mapped continuously onto $[0,1]^{ω_1}$. Also, without any additional axioms we prove that there is a non-separable growth of $ω$ supporting a measure and that there is a compactification $L$ of $ω$ with growth of such properties and such that the natural copy of $c_0$ is complemented in $C(L)$. Finally, we discuss examples of spaces not supporting measures but satisfying quite strong chain conditions. Our main tool is a characterization due to Kamburelis of Boolean algebras supporting measures in terms of their chain conditions in generic extensions by a measure algebra.

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arXiv2016

Measures and fibers

We study measures on compact spaces by analyzing the properties of fibers of continuous mappings into 2^omega. We show that if a compact zerodimensional space K carries a measure of uncountable Maharam type, then such a mapping has a non-scattered fiber and, if we assume additionally a weak version of Martin's Axiom, such a mapping has a fiber carrying a measure of uncountable Maharam type. Also, we prove that every compact zerodimensional space which supports a strictly positive measure and which can be mapped into 2^omega by a finite-to-one function is separable.

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arXiv2015

Measures on Suslinean spaces

We study the existence of non-separable compact spaces that support a measure and are small from the topological point of view. In particular, we show that under Martin's axiom there is a non-separable compact space supporting a measure which has countable $π$-character and which cannot be mapped continuously onto $[0,1]^{ω_1}$. On the other hand, we prove that in the random model there is no non-separable compact space having countable $π$-character and supporting a measure.

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arXiv2014

Hausdorff gaps and towers in P(ω)/Fin

We define and study two classes of uncountable $\subseteq^*$-chains: Hausdorff towers and Suslin towers. We discuss their existence in various models of set theory. Then, some of the results and methods are used to provide examples of indestructible gaps not equivalent to a Hausdorff gap. Also, we indicate possible ways of developing a structure theory for towers.

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arXiv2014

Representations of ideals in Polish groups and in Banach spaces

We investigate ideals of the form $\{A \subseteq ω\colon \sum_{n\in A} x_n$ is unconditionally convergent $\}$, where $(x_n)_{n\inω}$ is a sequence in a Polish group or in a Banach space. If an ideal on $ω$ can be seen in this form for some sequence in $X$, then we say that it is representable in $X$. After numerous examples we show the following theorems: (1) An ideal is representable in a Polish Abelian group iff it is an analytic P-ideal. (2) An ideal is representable in a Banach space iff it is a non-pathological analytic P-ideal. We focus on the family of ideals representable in $c_0$. We prove that the trace of the null ideal, Farah's ideal, and Tsirelson ideals are not representable in $c_0$, and that a tall $F_σ$ P-ideal is representable in $c_0$ iff it is a summable ideal. Also, we provide an example of a peculiar ideal which is representable in $\ell_1$ but not in $\mathbb{R}$. Finally, we summarize some open problems of this topic.

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