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Adam Kwela

Publications and source records attributed to Adam Kwela.

At least 19 recordsLinked to original sources
arXiv2026

Exh subspaces of bounded variation spaces generated by submeasures

In the present paper we are building one more bridge between the theory of ideals on the set of natural numbers and the theory of spaces of functions of bounded variation. It is already known that $F_σ$ ideals (that is, ideals of the form $\text{Fin}(ϕ)$, where $ϕ$ is a lower semicontinuous submeasure on $\mathbb{N}$) lead to a nice generalization of different concepts from the theory of functions of bounded variation (Waterman variation and Chanturya classes). In this paper we show that analytic P-ideals (that is, ideals of the form $\text{Exh}(ϕ)$, where $ϕ$ is a lower semicontinuous submeasure on $\mathbb{N}$) lead to a generalization of spaces of functions continuous in $Λ$-variation. The properties of this generalization are investigated.

math.FA↗
arXiv2026

A new order for ideal sequential compactness

Let $\mathcal{I}$ be an ideal on $ω$ and $X$ be a topological space. A sequence $(x_n)_{n\in ω}$ in $X$ is $\mathcal{I}$-convergent if there is $x\in X$ such that $\{n\in ω:x_n\notin U\}\in\mathcal{I}$ for every open neighborhood $U$ of $x$. We examine the following variant of sequential compactness associated with $\I$: $X$ is $\mathrm{BW}(\mathcal{I})$ if for every sequence $(x_n)_{n\in ω}$ in $X$ there is $A\notin\mathcal{I}$ such that $(x_n)_{n\in A}$ is $\mathcal{I}$-convergent. We introduce a new preorder on ideals, denoted $\leq_{BW}$, such that $\mathcal{I}\leq_{BW}\mathcal{J}$ implies that every $\mathrm{BW}(\mathcal{J})$ space is $\mathrm{BW}(\mathcal{I})$. Our main result states that under CH the above implication can be reversed in the case of $\mathbf{F_σ}$ ideals $\I$ and $\J$. We compare $\leq_{BW}$ with the Katětov order and study the relation $\leq_{BW}$ among some well-known ideals (e.g. the van der Waerden ideal $\mathcal{W}$ consisting of all subsets of $ω$ that do not contain arbitrary long finite arithmetic progressions). As a consequence, we answer two open questions posed by Filipów and Tryba in [Top. App. {\textbf{178}} (2014), 438--452] concerning comparison of $\mathrm{BW}(\mathcal{W})$ with the class of sequentially compact spaces.

math.GN↗
arXiv2026

Are scales Fréchet?

We continue the study of Dow spaces of a $\mathfrak{b}$-scale, originally introduced by Alan Dow in "$π$-Weight and the Fréchet-Urysohn property" (Topology and its Applications, Vol. 174, pp. 56-61). We prove that it is consistent that all such spaces are Fréchet, but it is also consistent that none of them is. We use these spaces to exhibit (consistently) a $\triangle_{2}^{1}$ ideal that does not satisfy the Category Dichotomy. Finally, we prove that the Category Dichotomy holds for all co-analytic ideals.

math.GN↗
arXiv2026

Critical partition regular functions for compact spaces

We study ideal-based refinements of sequential compactness arising from the class FinBW(I), consisting of topological spaces in which every sequence admits a convergent subsequence indexed by a set outside a given ideal I. A central theme of this work is the existence of critical ideals whose position in the Katetov order determines the relationship between a fixed class of spaces and the corresponding FinBW(I) classes. Building on earlier results characterizing several classical topological classes via such ideals, we extend this theory to a broader framework based on partition regular functions, which unifies ordinary convergence with other non-classical convergence notions such as IP- and Ramsey-type convergence. Furthermore, we investigate the existence of critical ideals associated with function classes motivated by Mazurkiewicz's theorem on uniformly convergent subsequences.

math.GN↗
arXiv2026

Sets of Ramsey-limit points and IP-limit points

Let $X$ be an uncountable Polish space and let $\mathcal{H}$ be the Hindman ideal, that is, the family of all $S\subseteq ω$ which are not $IP$-sets. For each sequence $x=(x_n)_{n \in ω}$ taking values in $X$, let $Λ_{x}(FS)$ be the set of $IP$-limit points of $x$. Also, let $Λ_{x}(\mathcal{H})$ be the set of $\mathcal{H}$-limit points of $x$, that is, the set of ordinary limits of subsequences $(x_n)_{n \in S}$ with $S\notin \mathcal{H}$. After proving that these two notions do not coincide in general, we show that both families of nonempty sets of the type $Λ_{x}(FS)$ and of the type $Λ_{x}(\mathcal{H})$ are precisely the class of nonempty analytic subsets of $X$. An analogous result holds also for Ramsey convergence. In the proofs, we use the concept of partition regular functions introduced in J. Symb. Log. (2024) [doi:10.1017/jsl.2024.8], which provide a unified approach to these types of convergence.

math.GN↗
arXiv2025

Zoo of ideal Schauder basis

We investigate the notion of filter (equivalently: ideal) Schauder basis of a Banach space. We do so by providing bunch of new examples of such bases that are not the standard ones, especially within classical Banach spaces ($\ell_p$, $c_0$, James' space). Those examples lead to distinguishing and characterizing filters (equivalently: ideals) in terms of Schauder bases. We investigate the relationship between possibly basic sequences and filters (equivalently: ideals) on the set of natural numbers.

math.FA↗
arXiv2025

Borel complexity of sets of ideal limit points

Let $X$ be an uncountable Polish space and let $\mathcal{I}$ be an ideal on $ω$. A point $η\in X$ is an $\mathcal{I}$-limit point of a sequence $(x_n)$ taking values in $X$ if there exists a subsequence $(x_{k_n})$ convergent to $η$ such that the set of indexes $\{k_n: n \in ω\}\notin \mathcal{I}$. Denote by $\mathscr{L}(\mathcal{I})$ the family of subsets $S\subseteq X$ such that $S$ is the set of $\mathcal{I}$-limit points of some sequence taking values in $X$ or $S$ is empty. In this paper, we study the relationships between the topological complexity of ideals $\mathcal{I}$, their combinatorial properties, and the families of sets $\mathscr{L}(\mathcal{I})$ which can be attained. On the positive side, we provide several purely combinatorial (not dependind on the space $X$) characterizations of ideals $\mathcal{I}$ for the inclusions and the equalities between $\mathscr{L}(\mathcal{I})$ and the Borel classes $Π^0_1$, $Σ^0_2$, and $Π^0_3$. As a consequence, we prove that if $\mathcal{I}$ is a $Π^0_4$ ideal then exactly one of the following cases holds: $\mathscr{L}(\mathcal{I})=Π^0_1$ or $\mathscr{L}(\mathcal{I})=Σ^0_2$ or $\mathscr{L}(\mathcal{I})=Σ^1_1$ (however we do not have an example of a $Π^0_4$ ideal with $\mathscr{L}(\mathcal{I})=Σ^1_1$). In addition, we provide an explicit example of a coanalytic ideal $\mathcal{I}$ for which $\mathscr{L}(\mathcal{I})=Σ^1_1$. On the negative side, we show that there are no ideals $\mathcal{I}$ such that $\mathscr{L}(\mathcal{I})=Π^0_2$ or $\mathscr{L}(\mathcal{I})=Σ^0_3$. We conclude with several open questions.

math.GN↗
arXiv2025

Critical ideals for compact spaces

For each countable ordinal $α$, we introduce an ideal $conv_α$ and use it to characterize the class of all compact countable spaces which are homeomorphic to the space $ω^α\cdot n+1$ with the order topology. The characterization is expressed in terms of finding a convergent subsequence defined on a set not belonging to $conv_α$.

math.GN↗
arXiv2025

Unboring ideals

Our main object of interest is the following notion: we say that a topological space space $X$ is in FinBW($\mathcal{I}$), where $\mathcal{I}$ is an ideal on $ω$, if for each sequence $(x_n)_{n\inω}$ in $X$ one can find an $A\notin\mathcal{I}$ such that $(x_n)_{n\in A}$ converges in $X$. We define an ideal $\mathcal{BI}$ which is critical for FinBW($\mathcal{I}$) in the following sense: Under CH, for every ideal $\mathcal{I}$, $\mathcal{BI}\not\leq_K\mathcal{I}$ ($\leq_K$ denotes the Katětov preorder of ideals) iff there is an uncountable separable space in FinBW($\mathcal{I}$). We show that $\mathcal{BI}\not\leq_K\mathcal{I}$ and $ω_1$ with the order topology is in FinBW($\mathcal{I}$), for all $\bf{Π^0_4}$ ideals $\mathcal{I}$. We examine when FinBW($\mathcal{I}$)$\setminus$FinBW($\mathcal{J}$) is nonempty: we prove under MA($σ$-centered) that for $\bf{Π^0_4}$ ideals $\mathcal{I}$ and $\mathcal{J}$ this is equivalent to $\mathcal{J}\not\leq_K\mathcal{I}$. Moreover, answering in negative a question of M. Hrušák and D. Meza-Alcántara, we show that the ideal $\text{Fin}\times\text{Fin}$ is not critical among Borel ideals for extendability to a $\bf{Π^0_3}$ ideal. Finally, we apply our results in studies of Hindman spaces and in the context of analytic P-ideals.

math.GN↗
arXiv2025

Inductive limits of ideals

G. Debs and J. Saint Raymond in 2009 defined the Borel separation rank of an analytic ideal $\mathcal{I}$ ($\text{rk}(\mathcal{I})$) as minimal ordinal $α<ω_{1}$ such that there is $\mathcal{S}\in\bf{Σ^0_{1+α}}$ with $\mathcal{I}\subseteq \mathcal{S}$ and $\mathcal{I}^\star\cap \mathcal{S}=\emptyset$, where $\mathcal{I}^\star$ is the filter dual to the ideal $\mathcal{I}$ (actually, the authors use the dual notion of filters instead of ideals). Moreover, they introduced ideals $\text{Fin}_α$, for all $α<ω_1$, and conjectured that $\text{rk}(\mathcal{I})\geqα$ if and only if $\mathcal{I}$ contains an isomorphic copy of $\text{Fin}_α$ ($\text{Fin}_α\sqsubseteq\mathcal{I}$). To define $\text{Fin}_α$ in the case of limit ordinals $0<α<ω_1$, G. Debs and J. Saint Raymond introduced inductive limits of ideals. We show that the above conjecture is false in the case of $α=ω$ by constructing an ideal $\text{Fin}'_ω$ of rank $ω$ such that $\text{Fin}_ω\not\sqsubseteq\text{Fin}'_ω$. However, we show that $\text{Fin}'_ω\sqsubseteq\mathcal{I}$ is equivalent to $\forall_{n\inω}\text{Fin}_n\sqsubseteq\mathcal{I}$. We discuss (indicated by the above result) possible modification of the original conjecture for limit ordinals.

math.LO↗
arXiv2025

On extendability to $F_σ$ ideals

Answering in negative a question of M. Hrušák, we construct a Borel ideal not extendable to any $F_σ$ ideal and such that it is not Katětov above the ideal $\mathrm{conv}$.

math.LO↗
arXiv2025

On a conjecture of Debs and Saint Raymond

Borel separation rank of an analytic ideal $\mathcal{I}$ on $ω$ is the minimal ordinal $α<ω_{1}$ such that there is $\mathcal{S}\in\bf{Σ^0_{1+α}}$ with $\mathcal{I}\subseteq \mathcal{S}$ and $\mathcal{I}^\star\cap \mathcal{S}=\emptyset$, where $\mathcal{I}^\star$ is the filter dual to the ideal $\mathcal{I}$. Answering in negative a question of G. Debs and J. Saint Raymond [Fund. Math. 204 (2009), no. 3], we construct a Borel ideal of rank $>2$ which does not contain an isomorphic copy of the ideal $\text{Fin}^3$.

math.LO↗
arXiv2025

Borel complexity of the family of attractors for weak IFSs

This paper is an attempt to measure the difference between the family of iterated function systems attractors and a broader family, the set of attractors for weak iterated function systems. We discuss Borel complexity of the set wIFS$^d$ of attractors for weak iterated function systems acting on $[0,1]^d$ (as a subset of the hyperspace $K([0,1]^d)$ of all compact subsets of $[0,1]^d$ equipped in the Hausdorff metric). We prove that wIFS$^d$ is $G_{δσ}$-hard in $K([0,1]^d)$, for all $d\in\mathbb{N}$. In particular, wIFS$^d$ is not $F_{σδ}$ (in contrast to the family IFS$^d$ of attractors for classical iterated function systems acting on $[0,1]^d$, which is $F_σ$). Moreover, we show that in the one-dimensional case, wIFS$^1$ is an analytic subset of $K([0,1])$.

math.DS↗
arXiv2025

Porosities of the sets of attractors

This paper is another attempt to measure the difference between the family $A[0,1]$ of attractors for iterated function systems acting on $[0,1]$ and a broader family, the set $A_w[0,1]$ of attractors for weak iterated function systems acting on $[0,1]$. It is known that both $A[0,1]$ and $A_w[0,1]$ are meager subsets of the hyperspace $K([0,1])$ (of all compact subsets of $[0,1]$ equipped in the Hausdorff metric). Actually, $A[0,1]$ is even $σ$-lower porous while the question about $σ$-lower porosity of $A_w[0,1]$ is still open. We prove that $A[0,1]$ is not $σ$-strongly porous in $K([0,1])$. Moreover, we show that $A_w[0,1]\setminus A[0,1]$ is dense in $K([0,1])$.

math.DS↗
arXiv2025

Egorov ideals

We study Egorov ideals, that is ideals on $ω$ for which the Egorov's theorem for ideal versions of pointwise and uniform convergences holds. We show that a non-pathological $\bf{Σ^0_2}$ ideal is Egorov if and only if it is countably generated. In particular, up to isomorphism, there are only three non-pathological $\bf{Σ^0_2}$ Egorov ideals. On the other hand, we construct $2^ω$ pairwise non-isomorphic Borel Egorov ideals. Moreover, we characterize when a product of ideals is Egorov.

math.LO↗
arXiv2025

Functions of bounded variation from ideal perspective

We present a unified approach to two classes of Banach spaces defined with the aid of variations: Waterman spaces and Chanturia classes. Our method is based on some ideas coming from the theory of ideals on the set of natural numbers.

math.FA↗
arXiv2025

Compactness in spaces of functions of bounded variation from ideal perspective

Recently we have presented a unified approach to two classes of Banach spaces defined by means of variations (Waterman spaces and Chanturia classes), utilizing the concepts from the theory of ideals on the set of natural numbers. We defined correspondence between an ideal on the set of natural numbers, a certain sequence space and related space of functions of bounded variation. In this paper, following these ideas, we give characterizations of compact embeddings between different Waterman spaces and between different Chanturia classes: both in terms of sequences defining these function spaces and in terms of properties of ideals corresponding to these function spaces.

math.FA↗