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Let (G,τ) be a Hausdorff Abelian topological group. It is called an s-group (resp. a bs-group) if there is a set S of sequences in G such that τ is the finest Hausdorff (resp. precompact) group topology on G in which every sequence of S converges to zero. Characterizations of Abelian s- and bs-groups are given. If (G,τ) is a maximally almost periodic (MAP) Abelian s-group, then its Pontryagin dual group ${(G,\tau )}^{\wedge }$ is a dense -closed subgroup of the compact group ${\left(G_d\right)}^{\wedge }$, where $G_d$ is the group G with the discrete...

It is well-known that every bounded Abelian group is a direct sum of finite cyclic subgroups. We characterize those non-trivial bounded subgroups $H$ of an infinite Abelian group $G$, for which there is an infinite subgroup $G_0$ of $G$ containing $H$ such that $G_0$ has a special decomposition into a direct sum which takes into account the properties of $G$, and which induces a natural decomposition of $H$ into a direct sum of finite subgroups.

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...

Following Banakh and Gabriyelyan (2016) we say that a Tychonoff space X is an Ascoli space if every compact subset of $C_k\left(X\right)$ is evenly continuous; this notion is closely related to the classical Ascoli theorem. Every $k_{ℝ}$-space, hence any k-space, is Ascoli. Let X be a metrizable space. We prove that the space $C_k\left(X\right)$ is Ascoli iff $C_k\left(X\right)$ is a $k_{ℝ}$-space iff X is locally compact. Moreover, $C_k\left(X\right)$ endowed with the weak topology is Ascoli iff X is countable and discrete. Using some basic concepts from probability theory and...