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The answer to this problem is negative: For the compact Polish group $X=S_3^\omega$ we have $Sn(X)\le\mathfrak r$ where $$\mathfrak r=\min\{|\mathcal R|:\mathcal R\subseteq [\omega]^\omega\;\wedge\;\f …

$\newcommand\Sn{\mathit{Sn}}$A subset $A$ of a group $X$ is called algebraic if $A=\{x\in X: a_0xa_1x\dotsm xa_n=1\}$ for some elements $a_0,a_1,\dotsc,a_n\in X$. Let $\mathcal A_X$ be the family of a …

$\DeclareMathOperator\cov{cov}$A subset $A$ of a semigroup $X$ is called algebraic if $$A=\{x\in X: a_0xa_1x...xa_n=b\}$$ for some $b\in X$ and $a_0,a_1,...,a_n \in X^1=X\cup \{1\}$. The smallest num …

A topological space $X$ is a $Q$-space if every subset of $X$ is of type $G_\delta$. The smallest cardinality of a metrizable separable space which is not a $Q$-space is denoted by $\mathfrak q_0$ and …

Let us recall that $\mathfrak b$ is the smallest cardinality of a subset of $\omega^\omega$, which cannot be covered by countably many compact subsets of $\omega^\omega$. The definition of $\mathfrak …

Let us recall that $\mathfrak t$ is the smallest cardinal $\kappa$ for which there exists a family $(T_\alpha)_{\alpha\in\kappa}$ of infinite subsets of $\omega$ such that $\bullet$ for any ordinals $ …

I am interested in a "dynamical" modification of the cardinals $\mathfrak r$ and $\mathfrak r_\sigma$, well-known in the theory of cardinal characteristics of the continuum. For a compact metrizable s …

Lyubomyr Zdomskyy proved that in the Laver model $\mathrm{cov}_H(2^\omega)=\omega_1<\mathfrak b=\mathfrak c$. His argument used the following known Laver property of the Laver model $V'$: for every fu …

Let $\mathrm{cov}_H(C_2^\omega)$ be the smallest cardinality of a cover of the Boolean group $C_2^\omega=(\mathbb Z/2\mathbb Z)^\omega$ by closed subgroups of infinite index. It can be shown that $$\m …

$\DeclareMathOperator\cov{cov}\newcommand\A{\text A}$A subset $A$ of a group $G$ is called affine if $A=xHy$ for some subgroup $H\subseteq G$ and some $x,y\in G$. Given a non-discrete topological grou …

For a subset $A\subseteq\mathbb T$ of the unit circle $\mathbb T=\{z\in\mathbb C:|z|=1\}$, let $\tau_A$ be the smallest topology on the additive group of integers $\mathbb Z$ such that for every $z\in …

The answer is affirmative and can be derived from Theorem (Banakh, Plichko). The Hilbert space $\ell_2(\aleph_1)$ condenses onto the Hilbert cube. By the way, this theorem is related to Problem 1 f …

Lyubomyr Zdomskyy informed me that the answer to my question is affirmative and follows from a recent (still unpublished) result of Osvaldo Guzman and Damjan Kalajdzievski who proved the consistency o …

A family $\mathcal A$ of infinite subsets of $\omega$ is called almost disjoint if for any distinct sets $A,B\in\mathcal A$ the intersection $A\cap B$ is finite. Let $\mathfrak a'$ be the largest car …

To my surprise, I found that this my ``new'' cardinal $\mathfrak{uf}$ is equal to $\mathfrak c$. Theorem. $\mathfrak{uf}=\mathfrak{c}$. Proof. Fix any maximal ultrafamily $\mathcal U\subseteq[\omeg …