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A function $f:X\to Y$ between topological spaces is called $\sigma$-continuous if there exists a countable cover $\mathcal C$ of $X$ such that for every $C\in\mathcal C$ the restriction $f{\restrictio …

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 …

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 $ …

Given two set $A,B$ we write $A\subset^* B$ if the complement $A\setminus B$ is infinite. A Hausdorff gap is a transfinite family $\langle A_\alpha,B_\alpha\rangle_{\alpha\in\omega_1}$ of infinite …

A tower is a subset $T\subset [\omega]^\omega$ of the family $[\omega]^\omega$ of all infinite subsets of $\omega$ such that $T$ is well-ordered by the relation $\supset^*$ of almost inclusion and has …

In this post I will discuss some cardinal characteristic of the continuum, related to partitions of $\omega$ and would like to know if it is equal to some known cardinal characteristic. By a partition …

A family $\mathcal U$ of infinite subsets of $\omega$ is called an ultrafamily if for any sets $U,V\in\mathcal U$ one of the sets $U\setminus V$, $U\cap V$ or $V\setminus U$ is finite. By the Kuratows …

Let $(A_\alpha)_{\alpha\in\mathfrak c}$ be an almost disjoint family of infinite subsets of $\omega$. The almost disjointness of the family means that $A_\alpha\cap A_\beta$ is finite for any ordinals …

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 …

Lyubomyr Zdomskyy informed me that the answer to my question is affirmative and follows from a recent (still unpublished) result of Osvaldo Guzman and Damjan Kalajdzievski who proved the consistency o …

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 …

This question was motivated by an answer to this question of Dominic van der Zypen. It relates to the following classical theorem of Sierpiński. Theorem (Sierpiński, 1921). For any countable partition …

Definition. A zero-dimensional topological space $X$ is called base zero-dimensional if for any base $\mathcal B$ of the topology that consists of closed-and-open sets in $X$, any open cover $\math …

A function $f:X\to Y$ is defined to be $\sigma$-continuous (resp. $\bar \sigma$-continuous) if there exists a countable (closed) cover $\mathcal C$ of $X$ such that the restriction $f{\restriction}C$ …

A function $f:\omega\to\omega$ is called $\bullet$ 2-to-1 if $|f^{-1}(y)|\le 2$ for any $y\in\omega$; $\bullet$ almost injective if the set $\{y\in \omega:|f^{-1}(y)|>1\}$ is finite. Let us introduce …