The following statement is proved to be independent from $[LB+\neg CH]$: $(*)$ Let $X$ be a Tychonoff space with $c\left(X\right)\le {\aleph }_0$ and $\pi w\left(X\right)<ℭ$. Then a union of less than $ℭ$ of nowhere dense subsets of $X$ is a union of not greater than $\pi w\left(X\right)$ of nowhere dense subsets.

We shall show that Open Coloring Axiom has different influence on the algebra $𝒫\left(\mathbb{N}\right)/fin$ than on ${\mathbb{N}}^{\mathbb{N}}/fin$. The tool used to accomplish this is forcing with a Suslin tree.

We study the deductive strength of properties under basic set-theoretical operations of the subclass E-Fin of the Dedekind finite sets in set theory without the Axiom of Choice ( AC ), which consists of all E-finite sets, where a set X is called E-finite if for no proper subset Y of X is there a surjection f:Y → X.

The Kalikow problem for a pair (λ,κ) of cardinal numbers,λ > κ (in particular κ = 2) is whether we can map the family of ω-sequences from λ to the family of ω-sequences from κ in a very continuous manner. Namely, we demand that for η,ν ∈ ω we have: η, ν are almost equal if and only if their images are. We show consistency of the negative answer, e.g., for ${\aleph }_{\omega }$ but we prove it for smaller cardinals. We indicate a close connection with the free subset property and its variants.

It was shown that there is a statistical learning problem – a version of the expectation maximization (EMX) problem – whose consistency in a domain of cardinality continuum under the family of purely atomic probability measures and with finite hypotheses is equivalent to a version of the continuum hypothesis, and thus independent of ZFC. K. P. Hart had subsequently proved that no solution to the EMX problem can be Borel measurable with regard to an uncountable standard Borel structure on $X$, and...

We show that in the ${\aleph }_2$-stage countable support iteration of Mathias forcing over a model of CH the complete Boolean algebra generated by absolutely divergent series under eventual dominance is not isomorphic to the completion of P(ω)/fin. This complements Vojtáš’ result that under $cf\left(\right)=$ the two algebras are isomorphic [15].

We prove that, under CH, for each Boolean algebra A of cardinality at most the continuum there is an embedding of A into P(ω)/fin such that each automorphism of A can be extended to an automorphism of P(ω)/fin. We also describe a model of ZFC + MA(σ-linked) in which the continuum is arbitrarily large and the above assertion holds true.

We show that for each natural number n > 1, it is consistent that there is a compact Hausdorff totally disconnected space $K_{2n}$ such that $C\left(K_{2n}\right)$ has no uncountable (semi)biorthogonal sequence ${(f_{\xi },{\mu }_{\xi })}_{\xi \in \omega ₁}$ where ${\mu }_{\xi }$’s are atomic measures with supports consisting of at most 2n-1 points of $K_{2n}$, but has biorthogonal systems ${(f_{\xi },{\mu }_{\xi })}_{\xi \in \omega ₁}$ where ${\mu }_{\xi }$’s are atomic measures with supports consisting of 2n points. This complements a result of Todorcevic which implies that it is consistent that such spaces do not exist: he proves that its is...

In set theory without the axiom of choice ($\mathrm{AC}$), we study certain non-constructive properties of infinite-dimensional vector spaces. Among several results, we establish the following: (i) None of the principles AC${}^{\mathrm{LO}}$ (AC for linearly ordered families of nonempty sets)—and hence AC${}^{\mathrm{WO}}$ (AC for well-ordered families of nonempty sets)— $\mathrm{DC}(<\kappa )$ (where $\kappa$ is an uncountable regular cardinal), and “for every infinite set $X$, there is a bijection $f:X\to \{0,1\}\times X$”, implies the statement “there exists a field $F$ such that every vector...

We prove that for an unbounded metric space $X$, the minimal character $𝗆\chi \left(\check{X}\right)$ of a point of the Higson corona $\check{X}$ of $X$ is equal to $𝔲$ if $X$ has asymptotically isolated balls and to $max\{𝔲,𝔡\}$ otherwise. This implies that under $𝔲<𝔡$ a metric space $X$ of bounded geometry is coarsely equivalent to the Cantor macro-cube $2^{<\mathbb{N}}$ if and only if $dim\left(\check{X}\right)=0$ and $𝗆\chi \left(\check{X}\right)=𝔡$. This contrasts with a result of Protasov saying that under CH the coronas of any two asymptotically zero-dimensional unbounded metric separable spaces are homeomorphic.

Given a compact Hausdorff space K we consider the Banach space of real continuous functions C(Kⁿ) or equivalently the n-fold injective tensor product $\otimes {̂}_{\epsilon }^{n}C\left(K\right)$ or the Banach space of vector valued continuous functions C(K,C(K,C(K...,C(K)...). We address the question of the existence of complemented copies of c₀(ω₁) in $\otimes {̂}_{\epsilon }^{n}C\left(K\right)$ under the hypothesis that C(K) contains such a copy. This is related to the results of E. Saab and P. Saab that $X\otimes {̂}_{\epsilon }Y$ contains a complemented copy of c₀ if one of the infinite-dimensional Banach...