Let a space $X$ be Tychonoff product ${\prod }_{\alpha <\tau }{X}_{\alpha }$ of $\tau$-many Tychonoff nonsingle point spaces $X_{\alpha }$. Let Suslin number of $X$ be strictly less than the cofinality of $\tau$. Then we show that every point of remainder is a non-normality point of its Čech–Stone compactification $\beta X$. In particular, this is true if $X$ is either $R^{\tau }$ or ${\omega }^{\tau }$ and a cardinal $\tau$ is infinite and not countably cofinal.

The notion of quasi-p-boundedness for p ∈ ${\omega }^{*}$ is introduced and investigated. We characterize quasi-p-pseudocompact subsets of β(ω) containing ω, and we show that the concepts of RK-compatible ultrafilter and P-point in ${\omega }^{*}$ can be defined in terms of quasi-p-pseudocompactness. For p ∈ ${\omega }^{*}$, we prove that a subset B of a space X is quasi-p-bounded in X if and only if B × $P_{RK}\left(p\right)$ is bounded in X × $P_{RK}\left(p\right)$, if and only if $c{l}_{\beta \left(X\times P_{RK}\left(p\right)\right)}\left(B\times P_{RK}\left(p\right)\right)=c{l}_{\beta X}B\times \beta \left(\omega \right)$, where $P_{RK}\left(p\right)$ is the set of Rudin-Keisler predecessors of p.

We discuss the following result of A. Szymański in “Retracts and non-normality points" (2012), Corollary 3.5.: If $F$ is a closed subspace of ${\omega }^{*}$ and the $\pi$-weight of $F$ is countable, then every nonisolated point of $F$ is a non-normality point of ${\omega }^{*}$. We obtain stronger results for all types of points, excluding the limits of countable discrete sets considered in “Some non-normal subspaces of the Čech–Stone compactification of a discrete space” (1980) by A. Błaszczyk and A. Szymański. Perhaps our proofs...

We consider discrete dynamical systems whose phase spaces are compact metrizable countable spaces. In the first part of the article, we study some properties that guarantee the continuity of all functions of the corresponding Ellis semigroup. For instance, if every accumulation point of $X$ is fixed, we give a necessary and sufficient condition on a point $a\in X^{\text{'}}$ in order that all functions of the Ellis semigroup $E(X,f)$ be continuous at the given point $a$. In the second part, we consider transitive dynamical...

We calculate the singular homology and Čech cohomology groups of the Harmonic Archipelago. As a corollary, we prove that this space is not homotopy equivalent to the Griffiths space. This is interesting in view of Eda’s proof that the first singular homology groups of these spaces are isomorphic.

We prove that if the Euclidean plane ${ℝ}^2$ contains an uncountable collection of pairwise disjoint copies of a tree-like continuum X, then the symmetric span of X is zero, sX = 0. We also construct a modification of the Oversteegen-Tymchatyn example: for each ε > 0 there exists a tree $X\subset {ℝ}^2$ such that σX < ε but X cannot be covered by any 1-chain. These are partial solutions of some well-known problems in continua theory.

We solve the long standing problem of characterizing the class of strongly Fréchet spaces whose product with every strongly Fréchet space is also Fréchet.

In the present article we provide an example of two closed non-$\sigma$-lower porous sets $A,B\subseteq ℝ$ such that the product $A\times B$ is lower porous. On the other hand, we prove the following: Let $X$ and $Y$ be topologically complete metric spaces, let $A\subseteq X$ be a non-$\sigma$-lower porous Suslin set and let $B\subseteq Y$ be a non-$\sigma$-porous Suslin set. Then the product $A\times B$ is non-$\sigma$-lower porous. We also provide a brief summary of some basic properties of lower porosity, including a simple characterization of Suslin non-$\sigma$-lower porous sets in topologically...

In this paper we construct a Kelley continuum $X$ such that $X\times [0,1]$ is not semi-Kelley, this answers a question posed by J.J. Charatonik and W.J. Charatonik in A weaker form of the property of Kelley, Topology Proc. 23 (1998), 69–99. In addition, we show that the hyperspace $C\left(X\right)$ is not semi- Kelley. Further we show that small Whitney levels in $C\left(X\right)$ are not semi-Kelley, answering a question posed by A. Illanes in Problemas propuestos para el taller de Teoría de continuos y sus hiperespacios, Queretaro, 2013.

∗ The first named author’s research was partially supported by GAUK grant no. 350, partially by the Italian CNR. Both supports are gratefully acknowledged. The second author was supported by funds of Italian Ministery of University and by funds of the University of Trieste (40% and 60%).Aiming to solve some open problems concerning pseudoradial spaces, we shall present the following: Assuming CH, there are two semiradial spaces without semi-radial product. A new property of pseudoradial spaces...