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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.

This is a continuation of the paper [J. Symb. Log. 87 (2022), 1065--1092]. For an ideal $\mathcal{I}$ on $\omega$ we denote $\mathcal{D}_{\mathcal{I}}=\{f\in\omega^\omega: f^{-1}[\{n\}]\in\mathcal{I} \text{ for every $n\in \omega$}\}$ and write $f\leq_{\mathcal{I}} g$ if $\{n\in\omega:f(n)>g(n)\}\in\mathcal{I}$, where $f,g\in\omega^\omega$. We study the cardinal numbers $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\cD_{\mathcal{I}} \times \cD_{\mathcal{I}}))$ describing the smallest sizes of subsets of $\mathcal{D}_{\mathcal{I}}$ that are unbounded from below with respect to $\leq_{\mathcal{I}}$. In particular, we examine the relationships of $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\cD_{\mathcal{I}} \times \cD_{\mathcal{I}}))$ with the dominating number $\mathfrak{d}$. We show that, consistently, $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\cD_{\mathcal{I}} \times \cD_{\mathcal{I}}))>\mathfrak{d}$ for some ideal $\mathcal{I}$, however $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\cD_{\mathcal{I}} \times \cD_{\mathcal{I}}))\leq\mathfrak{d}$ for all analytic ideals $\mathcal{I}$. Moreover, we give example of a Borel ideal with $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\cD_{\mathcal{I}} \times \cD_{\mathcal{I}}))=add(\mathcal{M})$.

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

We introduce a method that allows to turn topological questions about Hindman spaces into purely combinatorial questions about the Kat\v{e}tov order of ideals on $\mathbb{N}$. We also provide two applications of the method. (1) We characterize $F_\sigma$ ideals $\mathcal{I}$ for which there is a Hindman space which is not an $\mathcal{I}$-space under the continuum hypothesis. This reduces a topological question of Albin L. Jones about consistency of existence of a Hindman space which is not van der Waerden to the question whether the ideal of all non AP-sets is not below the ideal of all non IP-sets in the Kat\v{e}tov order. (2) Under the continuum hypothesis, we construct a Hindman space which is not an $\mathcal{I}_{1/n}$-space. This answers a question posed by Jana Fla\v{s}kov\'{a} at the 22nd Summer Conference on Topology and its Applications.

Following Baumgartner [J. Symb. Log. 60 (1995), no. 2], for an ideal $\mathcal{I}$ on $\omega$, we say that an ultrafilter $\mathcal{U}$ on $\omega$ is an $\mathcal{I}$-ultrafilter if for every function $f:\omega\to\omega$ there is $A\in \mathcal{U}$ with $f[A]\in \mathcal{I}$. If there is an $\mathcal{I}$-ultrafilter which is not a $\mathcal{J}$-ultrafilter, then $\mathcal{I}$ is not below $\mathcal{J}$ in the Kat\v{e}tov order $\leq_{K}$ (i.e. for every function $f:\omega\to\omega$ there is $A\in \mathcal{I}$ with $f^{-1}[A]\notin \mathcal{J}$). On the other hand, in general $\mathcal{I}\not\leq_{K}\mathcal{J}$ does not imply that existence of an $\mathcal{I}$-ultrafilter which is not a $\mathcal{J}$-ultrafilter is consistent. We provide some sufficient conditions on ideals to obtain the equivalence: $\mathcal{I}\not\leq_{K}\mathcal{J}$ if and only if it is consistent that there exists an $\mathcal{I}$-ultrafilter which is not a $\mathcal{J}$-ultrafilter. In some cases when the Kat\v{e}tov order is not enough for the above equivalence, we provide other conditions for which a similar equivalence holds. We are mainly interested in the cases when the family of all $\mathcal{I}$-ultrafilters or $\mathcal{J}$-ultrafilters coincides with some known family of ultrafilters: P-points, Q-points or selective ultrafilters (a.k.a. Ramsey ultrafilters). In particular, our results provide a characterization of Borel ideals $\mathcal{I}$ which can be used to characterize P-points as $\mathcal{I}$-ultrafilters. Moreover, we introduce a cardinal invariant which is used to obtain a sufficient condition for the existence of an $\mathcal{I}$-ultrafilter which is not a $\mathcal{I}$-ultrafilter. Finally, we prove some new results concerning existence of certain ultrafilters under various set-theoretic assumptions.

We examine topological spaces not distinguishing ideal pointwise and ideal $\sigma$-uniform convergence of sequences of real-valued continuous functions defined on them. For instance, we introduce a purely combinatorial cardinal characteristic (a sort of the bounding number $\mathfrak{b}$) and prove that it describes the minimal cardinality of topological spaces which distinguish ideal pointwise and ideal $\sigma$-uniform convergence. Moreover, we provide examples of topological spaces (focusing on subsets of reals) that do or do not distinguish the considered convergences. Since similar investigations for ideal quasi-normal convergence instead of ideal $\sigma$-uniform convergence have been performed in literature, we also study spaces not distinguishing ideal quasi-normal and ideal $\sigma$-uniform convergence of sequences of real-valued continuous functions defined on them.

Let $\mathcal{I}$ be an ideal on $\omega$. For $f,g\in\omega^\omega$ we write $f \leq_{\mathcal{I}} g$ if $f(n) \leq g(n)$ for all $n\in\omega\setminus A$ with some $A\in\mathcal{I}$. Moreover, we denote $\mathcal{D}_{\mathcal{I}}=\{f\in\omega^\omega: f^{-1}[\{n\}]\in\mathcal{I} \text{ for every $n\in \omega$}\}$ (in particular, $\mathcal{D}_{Fin}$ denotes the family of all finite-to-one functions). We examine cardinal numbers $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}}))$ and $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{Fin}\times \mathcal{D}_{Fin}))$ describing the smallest sizes of unbounded from below with respect to the order $\leq_{\mathcal{I}}$ sets in $\mathcal{D}_{Fin}$ and $\mathcal{D}_{\mathcal{I}}$, respectively. For a maximal ideal $\mathcal{I}$, these cardinals were investigated by M. Canjar in connection with coinitial and cofinal subsets of the ultrapowers. We show that $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{Fin} \times \mathcal{D}_{Fin})) =\mathfrak{b}$ for all ideals $\mathcal{I}$ with the Baire property and that $\aleph_1 \leq \mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}})) \leq\mathfrak{b}$ for all coanalytic weak P-ideals (this class contains all $\Pi^0_4$ ideals). What is more, we give examples of Borel (even $\Sigma^0_2$) ideals $\mathcal{I}$ with $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}}))=\mathfrak{b}$ as well as with $\mathfrak{b}(\geq_{\mathcal{I}}\cap (\mathcal{D}_{\mathcal{I}} \times \mathcal{D}_{\mathcal{I}})) =\aleph_1$.

For many years, there have been conducting research (e.g. by Bergelson, Furstenberg, Kojman, Kubi\'{s}, Shelah, Szeptycki, Weiss) into sequentially compact spaces that are, in a sense, topological counterparts of some combinatorial theorems, for instance Ramsey's theorem for coloring graphs, Hindman's finite sums theorem and van der Waerden's arithmetical progressions theorem. These spaces are defined with the aid of different kinds of convergences: IP-convergence, R-convergence and ordinary convergence. The first aim of this paper is to present a unified approach to these various types of convergences and spaces. Then, using this unified approach, we prove some general theorems about existence of the considered spaces and show that all results obtained so far in this subject can be derived from our theorems. The second aim of this paper is to obtain new results about the specific types of these spaces. For instance, we construct a Hausdorff Hindman space that is not an $\I_{1/n}$-space and a Hausdorff differentially compact space that is not Hindman. Moreover, we compare Ramsey spaces with other types of spaces. For instance, we construct a Ramsey space that is not Hindman and a Hindman space that is not Ramsey. The last aim of this paper is to provide a characterization that shows when there exists a space of one considered type that is not of the other kind. This characterization is expressed in purely combinatorial manner with the aid of the so-called Kat\v{e}tov order that has been extensively examined for many years so far. This paper may interest the general audience of mathematicians as the results we obtain are on the intersection of topology, combinatorics, set theory and number theory.

A family I of subsets of a set X is an ideal on X if it is closed under taking subsets and finite unions of its elements. An ideal I on X is below an ideal J on Y in the Katetov order if there is a function $f:Y\to X$ such that $f^{-1}[A]\in J$ for every $A\in I$. We show that the Hindman ideal, the Ramsey ideal and the summable ideal are pairwise incomparable in the Katetov order, where * the Ramsey ideal consists of those sets of pairs of natural numbers which do not contain a set of all pairs of any infinite set (equivalently do not contain, in a sense, any infinite complete subgraph), * the Hindman ideal consists of those sets of natural numbers which do not contain any infinite set together with all finite sums of its members (equivalently do not contain IP-sets that are considered in Ergodic Ramsey theory), * the summable ideal consists of those sets of natural numbers such that the series of the reciprocals of its members is convergent. Moreover, we show that in the Katetov order the above mentioned ideals are not below the van der Waerden ideal that consists of those sets of natural numbers which do not contain arithmetic progressions of arbitrary finite length.

We generalize the classical Olivier's theorem which says that for any convergent series $\sum_n a_n$ with positive nonincreasing real terms the sequence $(n a_n)$ tends to zero. Our results encompass many known generalizations of Olivier's theorem and give some new instances. The generalizations are done in two directions: we either drop the monotonicity assumption completely or we relax it to the monotonicity on a large set of indices. In both cases, the convergence of $(na_n)$ is replaced by ideal convergence. In the second part of the paper, we examine families of sequences for which the assertions of our generalizations of Olivier's theorem fail. Here, we are interested in finding large linear and algebraic substructures in these families.

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 $\sigma$-lower porous while the question about $\sigma$-lower porosity of $A_w[0,1]$ is still open. We prove that $A[0,1]$ is not $\sigma$-strongly porous in $K([0,1])$. Moreover, we show that $A_w[0,1]\setminus A[0,1]$ is dense in $K([0,1])$.

For $n,d\in\mathbb{N}$ we consider the families: - $L_n^d$ of attractors for iterated function systems (IFS) consisting of $n$ contractions acting on $[0,1]^d$, - $wL_n^d$ of attractors for weak iterated function systems (wIFS) consisting of $n$ weak contractions acting on $[0,1]^d$. We study closures of the above families as subsets of the hyperspace $K([0,1]^d)$ of all compact subsets of $[0,1]^d$ equipped in the Hausdorff metric. In particular, we show that $\overline{L_n^d}=\overline{wL_n^d}$ and $L_{n+1}^d\setminus\overline{L_n^d}\neq\emptyset$, for all $n,d\in\mathbb{N}$. What is more, we construct a compact set belonging to $\overline{L_2^d}$ which is not an attractor for any wIFS. We present a diagram summarizing our considerations.

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 $G_{\delta\sigma}$-hard in $K([0,1]^d)$, for all $d\in\mathbb{N}$. In particular, wIFS$^d$ is not $F_{\sigma\delta}$ (in contrast to the family IFS$^d$ of attractors for classical iterated function systems acting on $[0,1]^d$, which is $F_{\sigma}$). Moreover, we show that in the one-dimensional case, wIFS$^1$ is an analytic subset of $K([0,1])$.

Borel separation rank of an analytic ideal $\mathcal{I}$ on $\omega$ is the minimal ordinal $\alpha<\omega_{1}$ such that there is $\mathcal{S}\in\bf{\Sigma^0_{1+\alpha}}$ 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 $\text{Fin}^3$.

Answering in negative a question of M. Hru\v{s}\'ak, we construct a Borel ideal not extendable to any $F_\sigma$ ideal and such that it is not Kat\v{e}tov above the ideal $\mathrm{conv}$.

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 $\alpha<\omega_{1}$ such that there is $\mathcal{S}\in\bf{\Sigma^0_{1+\alpha}}$ 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 $\text{Fin}_\alpha$, for all $\alpha<\omega_1$, and conjectured that $\text{rk}(\mathcal{I})\geq\alpha$ if and only if $\mathcal{I}$ contains an isomorphic copy of $\text{Fin}_\alpha$ ($\text{Fin}_\alpha\sqsubseteq\mathcal{I}$). To define $\text{Fin}_\alpha$ in the case of limit ordinals $0<\alpha<\omega_1$, G. Debs and J. Saint Raymond introduced inductive limits of ideals. We show that the above conjecture is false in the case of $\alpha=\omega$ by constructing an ideal $\text{Fin}'_\omega$ of rank $\omega$ such that $\text{Fin}_\omega\not\sqsubseteq\text{Fin}'_\omega$. However, we show that $\text{Fin}'_\omega\sqsubseteq\mathcal{I}$ is equivalent to $\forall_{n\in\omega}\text{Fin}_n\sqsubseteq\mathcal{I}$. We discuss (indicated by the above result) possible modification of the original conjecture for limit ordinals.

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 $\omega$, if for each sequence $(x_n)_{n\in\omega}$ 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 Kat\v{e}tov 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 $\omega_1$ with the order topology is in FinBW($\mathcal{I}$), for all $\bf{\Pi^0_4}$ ideals $\mathcal{I}$. We examine when FinBW($\mathcal{I}$)$\setminus$FinBW($\mathcal{J}$) is nonempty: we prove under MA($\sigma$-centered) that for $\bf{\Pi^0_4}$ 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. Hru\v{s}\'ak and D. Meza-Alc\'antara, we show that the ideal $\text{Fin}\times\text{Fin}$ is not critical among Borel ideals for extendability to a $\bf{\Pi^0_3}$ ideal. Finally, we apply our results in studies of Hindman spaces and in the context of analytic P-ideals.

We show that there exist uncountably many (tall and nontall) pairwise nonisomorphic density-like ideals on $\omega$ which are not generalized density ideals. In addition, they are nonpathological. This answers a question posed by Borodulin-Nadzieja, Farkas, and Plebanek in [J. Symb. Log. \textbf{80} (2015), 1268--1289]. Lastly, we provide sufficient conditions for a density-like ideal to be necessarily a generalized density ideal., Comment: To appear in J. Symbolic Logic

One of the classical results concerning differentiability of continuous functions states that the set $\mathcal{SD}$ of somewhere differentiable functions (i.e., functions which are differentiable at some point) is Haar-null in the space $C[0,1]$. By a recent result of Banakh et al., a set is Haar-null provided that there is a Borel hull $B\supseteq A$ and a continuous map $f\colon \{0,1\}^\mathbb{N}\to C[0,1]$ such that $f^{-1}[B+h]$ is Lebesgue's null for all $h\in C[0,1]$. We prove that $\mathcal{SD}$ is not Haar-countable (i.e., does not satisfy the above property with "Lebesgue's null" replaced by "countable", or, equivalently, for each copy $C$ of $\{0,1\}^\mathbb{N}$ there is an $h\in C[0,1]$ such that $\mathcal{SD}\cap (C+h)$ is uncountable. Moreover, we use the above notions in further studies of differentiability of continuous functions. Namely, we consider functions differentiable on a set of positive Lebesgue's measure and functions differentiable almost everywhere with respect to Lebesgue's measure. Furthermore, we study multidimensional case, i.e., differentiability of continuous functions defined on $[0,1]^k$. Finally, we pose an open question concerning Takagi's function.

In this paper we are interested in the following notions of smallness: a subset $A$ of an abelian Polish group $X$ is called Haar-countable/Haar-finite/Haar-$n$ if there are a Borel hull $B\supseteq A$ and a copy $C$ of $2^\omega$ such that $(C+x)\cap B$ is countable/finite/of cardinality at most $n$, for all $x\in X$. Recently, Banakh et al. have unified the notions of Haar-null and Haar-meager sets by introducing Haar-$\mathcal{I}$ sets, where $\mathcal{I}$ is a collection of subsets of $2^\omega$. It turns out that if $\mathcal{I}$ is the $\sigma$-ideal of countable sets, the ideal of finite sets or the collection of sets of cardinality at most $n$, then we get the above notions. Moreover, those notions have been studied independently by Zakrzewski (under a different name -- perfectly $\kappa$-small sets). We study basic properties of the corresponding families of small sets, give suitable examples distinguishing them (in all abelian Polish groups of the form $\mathbb{R}\times X$) and study $\sigma$-ideals generated by compact members of the considered families. In particular, we show that Haar-countable sets do not form an ideal. Moreover, we answer some questions concerning null-finite sets, asked by Banakh and Jab{\l}o\'nska, and pose several open problems.

Let $G$ consist of all functions $g \colon \omega \to [0,\infty)$ with $g(n) \to \infty$ and $\frac{n}{g(n)} \nrightarrow 0$. Then for each $g\in G$ the family $\mathcal{Z}_g=\{A\subseteq\omega:\ \lim_{n\to\infty}\frac{\text{card}(A\cap n)}{g(n)}=0\}$ is an ideal associated to the notion of so-called upper density of weight $g$. Although those ideals have recently been extensively studied, they do not have their own name. In this paper, for Reader's convenience, we propose to call them simple density ideals. We show that there are $\mathfrak{c}$ many non-isomorphic (in fact even incomparable with respect to Kat\v{e}tov order) simple density ideals. Moreover, we prove that for a given $A\subset G$ with $\text{card}(A)<\mathfrak{b}$ one can construct a family of cardinality $\mathfrak{c}$ of pairwise incomparable (with respect to inclusion) simple density ideals which additionally are incomparable with all $\mathcal{Z}_g$ for $g\in A$. We show that this cannot be generalized to Kat\v{e}tov order as the ideal $\mathcal{Z}$ of sets of asymptotic density zero is maximal in the sense of Kat\v{e}tov order among all simple density ideals. We examine how many substantially different functions $g$ can generate the same ideal $\mathcal{Z}_g$ -- it turns out that the answer is either $1$ or $\mathfrak{c}$ (depending on $g$).

The main aim of this paper is to bridge two directions of research generalizing asymptotic density zero sets. This enables to transfer results concerning one direction to the other one. Consider a function $g\colon\omega\to [0,\infty)$ such that $\lim_{n\to\infty}g(n)=\infty$ and $\frac{n}{g(n)}$ does not converge to $0$. Then the family $\mathcal{Z}_g=\{A\subseteq\omega:\ \lim_{n\to\infty}\frac{\text{card}(A\cap n)}{g(n)}=0\}$ is an ideal called simple density ideal (or ideal associated to upper density of weight $g$). We compare this class of ideals with Erd\H{o}s-Ulam ideals. In particular, we show that there are $\sqsubseteq$-antichains of size $\mathfrak{c}$ among Erd\H{o}s-Ulam ideals which are and are not simple density ideals. We characterize simple density ideals which are Erd\H{o}s-Ulam as those containing the classical ideal of sets of asymptotic density zero. We also characterize Erd\H{o}s-Ulam ideals which are simple density ideals. In the latter case we need to introduce two new notions. One of them, called increasing-invariance of an ideal $\mathcal{I}$, asserts that given $B\in\mathcal{I}$ and $C\subseteq\omega$ with $\text{card}(C\cap n)\leq\text{card}(B\cap n)$ for all $n$, we have $C\in\mathcal{I}$. Finally, we pose some open problems.

This paper is devoted to studies of IwQN-spaces and some of their cardinal characteristics. Recently, \v{S}upina proved that I is not a weak P-ideal if and only if any topological space is an IQN-space. Moreover, under $\mathfrak{p}=\mathfrak{c}$ he constructed a maximal ideal I (which is not a weak P-ideal) for which the notions of IQN-space and QN-space do not coincide. In this paper we show that, consistently, there is an ideal I (which is not a weak P-ideal) for which the notions of IwQN-space and wQN-space do not coincide. We also prove that for this ideal the ideal version of Scheepers Conjecture does not hold (this is the first known example of such weak P-ideal). We obtain a strictly combinatorial characterization of ${\tt non}(\text{IwQN-space})$ similar to the one given by \v{S}upina in the case of ${\tt non}(\text{IQN-space})$. We calculate ${\tt non}(\text{IQN-space})$ and ${\tt non}(\text{IwQN-space})$ for some weak P-ideals. Namely, we show that $\mathfrak{b}\leq{\tt non}(\text{IQN-space})\leq{\tt non}(\text{IwQN-space})\leq\mathfrak{d}$ for every weak P-ideal I and that ${\tt non}(\text{IQN-space})={\tt non}(\text{IwQN-space})=\mathfrak{b}$ for every $\mathtt{F_\sigma}$ ideal I as well as for every analytic P-ideal I generated by an unbounded submeasure (this establishes some new bounds for $\mathfrak{b}(I,I,Fin)$). As a consequence, we obtain some bounds for ${\tt add}(\text{IQN-space})$. In particular, we get ${\tt add}(\text{IQN-space})=\mathfrak{b}$ for analytic P-ideals I generated by an unbounded submeasure. By a result of Bukovsk\'y, Das and \v{S}upina it is known that in the case of tall ideals I the notions of IQN-space (IwQN-space) and QN-space (wQN-space) cannot be distinguished. We prove that if I is a tall ideal and X is a topological space of cardinality less than ${\tt cov^*}(I)$, then X is an IwQN-space if and only if it is a wQN-space.

We say that an ideal I is homogeneous, if its restriction to any I-positive subset is isomorphic to I. The paper investigates basic properties of this notion -- we give examples of homogeneous ideals and present some applications to topology and ideal convergence. Moreover, we answer questions related to our research.

For any Borel ideal we characterize ideal equal Baire system generated by the families of continuous and quasi-continuous functions, i.e., the families of ideal equal limits of sequences of continuous and quasi-continuous functions.

We show that not every family of generalized microscopic sets forms an ideal. Moreover, we prove that some of these families have some weaker additivity properties and some of them do not have even that.

A set $M\subset\mathbb{R}$ is microscopic if for each $\varepsilon>0$ there is a sequence of intervals $(J_n)_{n\in\omega}$ covering $M$ and such that $|J_n|\leq \varepsilon^{n+1}$ for each $n\in\omega$. We show that there is a microscopic set which cannot be covered by a sequence $(J_n)_{n\in\omega}$ with $\{n\in\omega:J_n\neq\emptyset\}$ of lower asymptotic density zero. We prove (in ZFC) that additivity of the ideal of microscopic sets is $\omega_1$. This solves a problem of G. Horbaczewska. Finally, we discuss additivity of some generalizations of this ideal.

We give an exact value of the rank of an $\mathcal{F}$-Fubini sum of filters for the case where $\mathcal{F}$ is a Borel filter of rank $1$. We also consider $\mathcal{F}$-limits of filters $\mathcal{F}_i$, which are of the form $\lim_\mathcal{F}\mathcal{F}_i=\left\{A\subset X: \left\{i\in I: A\in\mathcal{F}_i\right\}\in\mathcal{F}\right\}$. We estimate the ranks of such filters; in particular we prove that they can fall to $1$ for $\mathcal{F}$ as well as for $\mathcal{F}_i$ of arbitrarily large ranks. At the end we prove some facts concerning filters of countable type and their ranks.

In this paper we study a new ideal $\mathcal{WR}$. The main result is the following: an ideal is not weakly Ramsey if and only if it is above $\mathcal{WR}$ in the Kat\v{e}tov order. Weak Ramseyness was introduced by Laflamme in order to characterize winning strategies in a certain game. We apply result of Natkaniec and Szuca to conclude that $\mathcal{WR}$ is critical for ideal convergence of sequences of quasi-continuous functions. We study further combinatorial properties of $\mathcal{WR}$ and weak Ramseyness. Answering a question of Filip\'ow et al. we show that $\mathcal{WR}$ is not $2$-Ramsey, but every ideal on $\omega$ isomorphic to $\mathcal{WR}$ is Mon (every sequence of reals contains a monotone subsequence indexed by a $\mathcal{I}$-positive set).

This paper studies the combinatorics of ideals which recently appeared in ergodicity results for analytic equivalence relations. The ideals have the following topological representation. There is a separable metrizable space $X$, a $\sigma$-ideal $I$ on $X$ and a dense countable subset $D$ of $X$ such that the ideal consists of those subsets of $D$ whose closure belongs to $I$. It turns out that this definition is indepedent of the choice of $D$. We show that an ideal is of this form if and only if it is dense and countably separated. The latter is a variation of a notion introduced by Todor\vcevi\'c for gaps. As a corollary, we get that this class is invariant under the Rudin--Blass equivalence. This also implies that the space $X$ can be always chosen to be compact so that $I$ is a $\sigma$-ideal of compact sets. We compute the possible descriptive complexities of such ideals and conclude that all analytic equivalence relations induced by such ideals are $\mathbf{\Pi}^0_3$. We also prove that a coanalytic ideal is an intersection of ideals of this form if and only if it is weakly selective.