On topological properties of the group of the null sequences valued in an Abelian topological group

Following [23], denote by ${\mathfrak{F}}_0$ the functor on the category $\mathbf{T}\mathbf{A}\mathbf{G}$ of all Hausdorff Abelian topological groups and continuous homomorphisms which passes each $X\in \mathbf{T}\mathbf{A}\mathbf{G}$ to the group of all $X$-valued null sequences endowed with the uniform topology. We prove that if $X\in \mathbf{T}\mathbf{A}\mathbf{G}$ 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) $\mathfrak{F}_0 (X)$isaSchwartzgroup,3)$\mathfrak{F}_0 (X)$respectscompactness,4)$\mathfrak{F}_0(X)$hastheSchurproperty\mathrm{.}So,ifaLCAgroup$X$ 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 $X=\mathbb{T}$.