arXiv subjects

Jaroslav Supina

Publications and source records attributed to Jaroslav Supina.

5 recordsLinked to original sources

arXiv2025

Cardinal characteristics associated with small subsets of reals

Inspired by Bartoszyński's work on small sets, we introduce a new ideal defined by interval partitions on natural numbers and summable sequences of positive reals. Similarly, we present another ideal that relies on Bartoszyński's and Shelah's representation of $F_σ$ measure zero sets. We show they are $σ$-ideals characterizing all small sets and $F_σ$ measure zero sets. We also study the cardinal characteristics associated with the introduced ideals. We use them to describe the invariants of measure, discuss their connection to Cichoń's diagram, and present related consistency results.

math.LO↗

arXiv2024

Slalom numbers

The paper is an extensive and systematic study of cardinal invariants we call slalom numbers, describing the combinatorics of sequences of sets of natural numbers. Our general approach, based on relational systems, covers many such cardinal characteristics, including localization and anti-localization cardinals. We show that most of the slalom numbers are connected to topological selection principles, in particular, we obtain the representation of the uniformity of meager and the cofinality of measure. Considering instances of slalom numbers parametrized by ideals on natural numbers, we focus on monotonicity properties with respect to ideal orderings and computational formulas for the disjoint sum of ideals. Hence, we get such formulas for several pseudo-intersection numbers as well as for the bounding and dominating numbers parametrized with ideals. Based on the effect of adding a Cohen real, we get many consistent constellations of different values of slalom numbers.

math.LO↗

arXiv2024

Open filters and measurable cardinals

In this paper, we investigate the poset $\mathbf{OF}(X)$ of free open filters on a given space $X$. In particular, we characterize spaces for which $\mathbf{OF}(X)$ is a lattice. For each $n\in\mathbb{N}$ we construct a scattered space $X$ such that $\mathbf{OF}(X)$ is order isomorphic to the $n$-element chain, which implies the affirmative answer to two questions of Mooney. Assuming CH we construct a scattered space $X$ such that $\mathbf{OF}(X)$ is order isomorphic to $(ω+1,\geq)$. To prove the latter facts we introduce and investigate a new stratification of ultrafilters which depends on scattered subspaces of $β(κ)$. Assuming the existence of $n$ measurable cardinals, for every $m_0,\ldots,m_{n}\in\mathbb N$ we construct a space $X$ such that $\mathbf{OF}(X)$ is order isomorphic to $\prod_{i=0}^nm_i$. Also, we show that the existence of a metric space possessing a free $ω_1$-complete closed, $G_δ$, $F_σ$ or Borel ultrafilter is equivalent to the existence of a measurable cardinal.

math.GN↗

arXiv2023

On P-like ideals induced by disjoint families

We consider a combinatorial property isolated in the field of ideal convergence, a P-property for two ideals on natural numbers. We show that among selected ideals induced by disjoint families, not all pairs satisfy P-property for two ideals. In many cases we specify the inducing partitions for which the corresponding ideals possess the property. Regarding selector ideals, a useful coloring-like necessary condition is provided.

math.GN↗

arXiv2023

Ideal approach to convergence in functional spaces

We solve the last standing open problem from the seminal paper by J. Gerlits and Zs. Nagy, which was later reposed by A. Miller, T. Orenshtein and B. Tsaban. Namely, we show that under p = c there is a δ-set that is not a γ-set. Thus we construct a set of reals A such that the space Cp(A) of all real-valued continuous functions on A is not Frechet-Urysohn, but possesses the Pytkeev property. Moreover, under CH we construct a π-set that is not a δ-set solving a problem by M. Sakai. In fact, we construct various examples of δ-sets that are not γ-sets, satisfying finer properties parametrized by ideals on natural numbers. Finally, we distinguish ideal variants of the Frechet-Urysohn property for many different Borel ideals in the realm of functional spaces.

math.GN↗