@techreport{bfa05d325c0e4bd49499bb9639bf27be,

title = "On topological properties of the group of the null sequences valued in an Abelian topological group",

abstract = " Following [23], denote by $\mathfrak{F}_0$ the functor on the category $\mathbf{TAG}$ of all Hausdorff Abelian topological groups and continuous homomorphisms which passes each $X\in \mathbf{TAG}$ to the group of all $X$-valued null sequences endowed with the uniform topology. We prove that if $X\in \mathbf{TAG}$ is an $(E)$-space (respectively, a strictly angelic space or a \v{S}-space), then $\mathfrak{F}_0 (X)$ is an $(E)$-space (respectively, a strictly angelic space or a \v{S}-space). We study respected properties for topological groups in particular from categorical point of view. Using this investigation we show that for a locally compact Abelian (LCA) group $X$ the following are equivalent: 1) $X$ is totally disconnected, 2) $\mathfrak{F}_0 (X)$ is a Schwartz group, 3) $\mathfrak{F}_0 (X)$ respects compactness, 4) $\mathfrak{F}_0(X)$ has the Schur property. So, if a LCA group $X$ has non-zero connected component, the group $\mathfrak{F}_0(X)$ is a reflexive non-Schwartz group which does not have the Schur property. We prove also that for every compact connected metrizable Abelian group $X$ the group $\mathfrak{F}_0 (X)$ is monothetic that generalizes a result by Rolewicz for $X=\mathbb{T}$. ",

keywords = "math.GR, 22A10, 22A35, 43A05, 43A40, 54H11",

author = "Gabriyelyan, {S. S.}",

year = "2013",

language = "???core.languages.en_GB???",

series = "Arxiv preprint",

edition = " arXiv:1306.5117 [math.GR]",

type = "WorkingPaper",

}