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The answer is strong YES for connected spaces admitting a non-constant continuous function and NO in the opposite case. If $X$ is a connected topological space that admits a non-constant continuous …

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 …

Let $\mathcal F$ be a free filter on $\omega$ and $$\mathcal F^+:=\{E\subset \omega:\forall F\in\mathcal F\;E\cap F\ne\emptyset\}.$$ A family $\mathcal N$ of subsets of $\omega$ is called a network fo …

A group $G$ is defined to have the Bergman property if for any subset $X=X^{-1}$ generating $G$ there exists $n$ such that $X^n=G$. By a result of Bergman, the permutation group of any set has the B …

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 $V:=\omega\times\omega$ and $E=\{E_{n,m}:n,m\in\omega\}$ where $$E_{n,m}:=(\{0,\dots,n\}\times\{m\})\cup\{(2n,m+1)\}.$$ It seems that the hypergraph $(V,E)$ has no minimal cover. A simplification …

Maybe I posed this question too quickly: for the 6 hours that passed since the time of asking this question I have found a (relatively simple) counterexample to my Problem 2. Example. There exists a …

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 …

The answer to this question is "No". A non-affine biaffine Cartesian plane can be constructed as follows. First, we fix a suitable terminology. Every function $F:X\to Y$ is identified with its graph …

This question concerns the (synthetic) geometry of linear spaces. Definition 1. A linear space is a pair $(P,\mathcal L)$ consisting of a set $P$ whose elements are called points and a family $\mathca …

Let $A,B$ be two uncountable sets in a group $G$ such that for any elements $x,y\in G$ the intersection $xA\cap yB$ is finite. Let $\Phi:G\to 2^G$ be a function assigning to each element $x\in G$ some …

Problem 1. Is it true that each uncountable group $G$ contains two subsets $A,B\subset G$ such that 1) for any $x,y\in G$ the intersection $xA\cap yB$ is finite and 2) for any function $ …

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 …

1) This question (as posed) has a simple negative answer. Just take any countable set $x$ and put $M$ be the family of finite subsets of $x$. Then for the family $\mathcal A$ of all possible partition …

Unfortunately (for my further plans) this question has negative answer. Just take any two disjoint uncountable sets $A,B$ and consider the free group $G$ over the union $A\cup B$. Let $\Phi:G\to 2^G$ …