A compact topological space K is in the class A if it is homeomorphic to a subspace H of [0,1]I, for some set of indexes I, such that, if L is the subset of H consisting of all {xi : i C I} with xi=0 except for a countable number of i's, then L is dense in H. In this paper we show that the class A of compact spaces is not stable under continuous maps. This solves a problem posed by Deville, Godefroy and Zizler.

Let G be a locally compact group with a fixed left Haar measure. Given Young functions φ and ψ, we consider the Orlicz spaces $L^{\phi }\left(G\right)$ and $L^{\psi }\left(G\right)$ on a non-unimodular group G, and, among other things, we prove that under mild conditions on φ and ψ, the set $(f,g)\in L^{\phi }\left(G\right)\times L^{\psi }\left(G\right):f*g$ is well defined on G is σ-c-lower porous in $L^{\phi }\left(G\right)\times L^{\psi }\left(G\right)$. This answers a question raised by Głąb and Strobin in 2010 in a more general setting of Orlicz spaces. We also prove a similar result for non-compact locally compact groups.

Mappings preserving Cauchy sequences and certain types of convergences connected with these mappings are investigated.

A scadic space is a Hausdorff continuous image of a product of compact scattered spaces. We complete a theorem begun by G. Chertanov that will establish that for each scadic space X, χ(X) = w(X). A ξ-adic space is a Hausdorff continuous image of a product of compact ordinal spaces. We introduce an either-or chain condition called Property $R_{\lambda }^{\text{'}}$ which we show is satisfied by all ξ-adic spaces. Whereas Property $R_{\lambda }^{\text{'}}$ is productive, we show that a weaker (but more natural) Property $R_{\lambda }$ is not productive. Polyadic...

We prove that for an unbounded metric space $X$, the minimal character $𝗆\chi \left(\check{X}\right)$ of a point of the Higson corona $\check{X}$ of $X$ is equal to $𝔲$ if $X$ has asymptotically isolated balls and to $max\{𝔲,𝔡\}$ otherwise. This implies that under $𝔲<𝔡$ a metric space $X$ of bounded geometry is coarsely equivalent to the Cantor macro-cube $2^{<\mathbb{N}}$ if and only if $dim\left(\check{X}\right)=0$ and $𝗆\chi \left(\check{X}\right)=𝔡$. This contrasts with a result of Protasov saying that under CH the coronas of any two asymptotically zero-dimensional unbounded metric separable spaces are homeomorphic.

Let $X$ be an Abelian topological group. A subgroup $H$ of $X$ is characterized if there is a sequence $𝐮=\left\{u_n\right\}$ in the dual group of $X$ such that $H=\{x\in X:(u_n,x)\to 1\}$. We reduce the study of characterized subgroups of $X$ to the study of characterized subgroups of compact metrizable Abelian groups. Let $c_0\left(X\right)$ be the group of all $X$-valued null sequences and ${𝔲}_0$ be the uniform topology on $c_0\left(X\right)$. If $X$ is compact we prove that $c_0\left(X\right)$ is a characterized subgroup of $X^{\mathbb{N}}$ if and only if $X\cong {𝕋}^n\times F$, where $n\ge 0$ and $F$ is a finite Abelian group. For every compact Abelian...