We analyze the structure of strongly dominating sets of reals introduced in Goldstern et al. (Proc Am Math Soc 123(5):1573–1581, 1995). We prove that for every κ (...) class l of Baire sets for the Laver category base and compare the σ-ideal of sets which are not strongly dominating with the Laver ideal l0. (shrink)

We define the ideal with the property that a real omits all Borel sets in the ideal which are coded in a transitive model if and only if it is an amoeba real over this model. We investigate some other properties of this ideal. Strolling through the "amoeba forest" we gain as an application a modification of the proof of the inequality between the additivities of Lebesgue measure and Baire category.

We define a σ-ideal J D on the set of functions ω ω with the property that a real x ∈ ω ω is a Hechler real over V if and only if x omits all Borel sets in J D . In fact we define a topology D on ω ω related to Hechler forcing such that J D is the family of first category sets in D. We study cardinal invariants of the ideal J D.

We study cardinal invariants of systems of meager hereditary families of subsets of ω connected with the collapse of the continuum by Sacks forcing S and we obtain a cardinal invariant yω such that S collapses the continuum to yω and y ≤ yω ≤ b. Applying the Baumgartner-Dordal theorem on preservation of eventually narrow sequences we obtain the consistency of y = yω < b. We define two relations $\leq _{0}^{\ast}$ and $\leq _{1}^{\ast}$ on the set $(^{\omega}\omega)_{{\rm Fin}}$ of (...) finite-to-one functions which are Tukey equivalent to the eventual dominance relation of functions such that if $\germ{F}\subseteq (^{\omega}\omega)_{Fin}$ is $\leq _{1}^{\ast}$ -unbounded, well-ordered by $\leq _{1}^{\ast}$ , and not $\leq _{0}^{\ast}$ -dominating, then there is a nonmeager p-ideal. The existence of such a system F follows from Martin's axiom. This is an analogue of the results of [3], [9, 10] for increasing functions. (shrink)

We investigate the classes of ideals for which the Egoroff’s theorem or the generalized Egoroff’s theorem holds between ideal versions of pointwise and uniform convergences. The paper is motivated by considerations of Korch :269–282, 2017).

We continue the study of Rosenthal families initiated by Damian Sobota. We show that every Rosenthal filter is the intersection of a finite family of ultrafilters that are pairwise incomparable in the Rudin-Keisler partial ordering of ultrafilters. We introduce a property of filters, called an \-filter, properly between a selective filter and a \-filter. We prove that every \-ultrafilter is a Rosenthal family. We prove that it is consistent with ZFC to have uncountably many \-ultrafilters such that any intersection of (...) finitely many of them is a Rosenthal filter. (shrink)

We study the sets of symmetric continuity of real functions in connection with the sets of continuity. We prove that sets of reals of cardinality (...) of symmetric continuity of a real function that are not continuity points does not contain a nonmeager set with Baire property and has inner measure zero by introducing another notion of smallness below meager and measure zero. (shrink)