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There is a counterexample as a subset of the real plane endowed with the natural partial order. Take the graph $\Gamma$ of the standard continuous monotone map $f:C\to[0,1]$ of the Cantor set $C\subs …

First note that a topological space is zdH if and only if it is totally disconnected. Theorem 1. Each minimal totally disconnected topological space $X$ is compact. Proof: The total disconnectedness …

For a set $X$ we endow the set $\omega^X$ of all functions from $X$ to $\omega$ with the natural partial order $\le$ defined by $f\le g$ iff $f(x)\le g(x)$ for all $x\in X$. A function $f:\omega^\ome …

The answer to this problem is negative and follows from Theorem. For any monotone function $\mu:\omega^\omega\to\omega^{\omega_1}$ there exists a countable infinite set $A\subset\omega_1$ such that f …

For a set $X$ we endow the set $\omega^X$ of all functions from $X$ to $\omega$ with the natural partial order $\le$ defined by $f\le g$ iff $f(x)\le g(x)$ for all $x\in X$. A function $\mu:\omega^\om …

First some definitions. A semilattice is a commutative semigroup consisting of idempotents (i.e., elements such that $xx=x$). A typical example of a semilattice is the unit interval endowed with the …

For a countable space $X$ the discussed property means that $X$ is homeomorphic to $X/A$ for any finite subset $A\subset X$. Let us call this property quotient-homogeneous. There are many quotient-h …

It seems that the answer to this problem is affirmative: Take a well-behaved pospace $P$. To show that $\beta_2(P)$ is well-behaved, take any clopen upper set $U\subset \beta_2(P)$. We should prove t …

A positive answer to this problem is given by the known answers to the following problem posed by de Groot in New Scottish book. Problem 393 (de Groot; 28 May, 1958). Does there exist a (plane) conti …

Theorem. The topological sum $X=\bigoplus_{n\in\omega}\ell_2(\aleph_n)$ of Hilbert spaces of density $\aleph_n$ does not contain maximal homogeneous subspaces. Proof. Let $H$ be a non-empty homogeneo …

The answer to this question is negative. As a counterexample, consider the one-point extension $P:=2^{<\omega}\cup\{\infty\}$ of the binary tree $2^{<\omega}=\bigcup_{n\in\omega}2^n$. Here $2$ is the …

It is well known that the sum $\alpha+\beta$ of two ordinals $\alpha,\beta$ can be defined "geometrically" as the order type of the sum $(\{0\}\times \alpha)\cup(\{1\}\times\beta)$ endowed with the le …

For a cardinal $\kappa$ let $[\kappa]^{<\kappa}$ denote the family of subsets of cardinality $<\kappa$ in $\kappa$. The family $[\kappa]^{<\kappa}$ is endowed with the partial order of inclusion. A si …

A cover $\mathcal C$ of a set $X$ by subsets of $X$ is called $\bullet$ minimal if for every $C\in\mathcal C$ the family $\mathcal C\setminus\{C\}$ is not a cover of $X$; $\bullet$ minimizable if $\ …

The answer to this question is affirmative: Theorem. There exists a countable set $X$ and an uncountable family $\mathcal F$ of self-functions of $X$ such that the poset $T_2(\mathcal F)$ has no min …