We introduce the properties of a space to be strictly $WFU\left(M\right)$ or strictly $SFU\left(M\right)$, where $\varnothing \ne M\subset {\omega }^{*}$, and we analyze them and other generalizations of $p$-sequentiality ($p\in {\omega }^{*}$) in Function Spaces, such as Kombarov’s weakly and strongly $M$-sequentiality, and Kocinac’s $WFU\left(M\right)$ and $SFU\left(M\right)$-properties. We characterize these in $C_{\pi }\left(X\right)$ in terms of cover-properties in $X$; and we prove that weak $M$-sequentiality is equivalent to $WFU\left(L\right(M\left)\right)$-property, where $L\left(M\right)=\{{}^{\lambda }p:\lambda <{\omega }_1$ and $p\in M\}$, in the class of spaces which are $p$-compact for every $p\in M\subset {\omega }^{*}$; and that $C_{\pi }\left(X\right)$ is a $WFU\left(L\right(M\left)\right)$-space iff $X$ satisfies...

Bi-capacities have been recently introduced as a natural generalization of capacities (or fuzzy measures) when the underlying scale is bipolar. They allow to build more flexible models in decision making, although their complexity is of order $3^n$, instead of $2^n$ for fuzzy measures. In order to reduce the complexity, the paper proposes the notion of $p$-symmetric bi- capacities, in the same spirit as for $p$-symmetric fuzzy measures. The main idea is to partition the set of criteria (or states of nature,...

We parametrize Cichoń’s diagram and show how cardinals from Cichoń’s diagram yield classes of small sets of reals. For instance, we show that there exist subsets N and M of $w^w\times 2^w$ and continuous functions $e,f:w^w\to w^w$ such that  • N is $G_{\delta }$ and $N_x:x\in w^w$, the collection of all vertical sections of N, is a basis for the ideal of measure zero subsets of $2^w$;  • M is $F_{\sigma }$ and $M_x:x\in w^w$ is a basis for the ideal of meager subsets of $2^w$;  •$\forall x,y{N}_{e\left(x\right)}\subseteq N_y\Rightarrow M_x\subseteq M_{f\left(y\right)}$. From this we derive that for a separable metric space X,  •if for all Borel (resp. $G_{\delta }$) sets $B\subseteq X\times 2^w$ with all...

Let m ≥ 2 be an integer. We show that ZF + “Every countable set of m-element sets has an infinite partial choice function” is not strong enough to prove that every countable set of m-element sets has a choice function, answering an open question from . (Actually a slightly stronger result is obtained.) The independence result in the case where m = p is prime is obtained by way of a permutation (Fraenkel-Mostowski) model of ZFA, in which the set of atoms (urelements) has the structure of a vector...

We prove the results stated in the title.

Motivated by an application to the unconditional basic sequence problem appearing in our previous paper, we introduce analogues of the Laver ideal on ℵ₂ living on index sets of the form ${\left[{\aleph }_k\right]}^{\omega }$ and use this to refine the well-known high-dimensional polarized partition relation for ${\aleph }_{\omega }$ of Shelah.

Let κ > ω be a regular cardinal and λ > κ a cardinal. The following partition property is shown to be consistent relative to a supercompact cardinal: For any $f:{\cup }_{n<\omega }{\left[X\right]}_{\subset }^n\to \gamma$ with $X\subset P_{\kappa }\lambda$ unbounded and 1 < γ < κ there is an unbounded Y ∪ X with $|f^{\text{'}\text{'}}{\left[Y\right]}_{\subset }^n|=1$ for any n < ω.

A combinatorial statement concerning ideals of countable subsets of ω is introduced and proved to be consistent with the Continuum Hypothesis. This statement implies the Suslin Hypothesis, that all (ω, ω*)-gaps are Hausdorff, and that every coherent sequence on ω either almost includes or is orthogonal to some uncountable subset of ω.

We investigate whether an arbitrary base for a dense-in-itself topological space can be partitioned into two bases. We prove that every base for a $T_3$ Lindelöf topology can be partitioned into two bases while there exists a consistent example of a first-countable, 0-dimensional, Hausdorff space of size $2^{\omega }$ and weight ${\omega }_1$ which admits a point countable base without a partition to two bases.

Three sets occurring in functional analysis are shown to be of class PCA (also called ${\Sigma }_2^1$) and to be exactly of that class. The definition of each set is close to the usual objects of modern analysis, but some subtlety causes the sets to have a greater complexity than expected. Recent work in a similar direction is in [1, 2, 10, 11, 12].

The additivity spectrum $ADD\left(ℐ\right)$ of an ideal $ℐ\subset 𝒫\left(I\right)$ is the set of all regular cardinals $\kappa$ such that there is an increasing chain $\{A_{\alpha }:\alpha <\kappa \}\subset ℐ$ with ${\bigcup }_{\alpha <\kappa }{A}_{\alpha }\notin ℐ$. We investigate which set $A$ of regular cardinals can be the additivity spectrum of certain ideals. Assume that $ℐ=ℬ$ or $ℐ=𝒩$, where $ℬ$ denotes the $\sigma$-ideal generated by the compact subsets of the Baire space ${\omega }^{\omega }$, and $𝒩$ is the ideal of the null sets. We show that if $A$ is a non-empty progressive set of uncountable regular cardinals and $pcf\left(A\right)=A$, then $ADD\left(ℐ\right)=A$ in some c.c.c generic extension of the...

We study several perfect set properties of the Baire space which follow from the Ramsey property $\omega \to {\left(\omega \right)}^{\omega }$. In particular we present some independence results which complete the picture of how these perfect set properties relate to each other.

We study splitting, infinitely often equal (ioe) and refining families from the descriptive point of view, i.e. we try to characterize closed, Borel or analytic such families by proving perfect set theorems. We succeed for $G_{\delta }$ hereditary splitting families and for analytic countably ioe families. We construct several examples of small closed ioe and refining families.