关于阿贝尔拓扑群的零序列群的拓扑性质
On topological properties of the group of the null sequences valued in an Abelian topological group
摘要：
在[23]之后，用$\mathfrak{F}_0$表示所有Hausdorff Abel拓扑群和连续同态的范畴$\mathbf{TAG}$上的函子，该函子将每个$X\in \mathbf{TAG}$传递到赋予一致拓扑的所有$X$值空序列的群。证明了若$X\in \mathbf{TAG}$是$(E)$ -空间（分别是严格天使空间或严格空间），则$\mathfrak{F}_0 (X)$是$(E)$ -空间（分别是严格天使空间或严格空间）.我们特别从范畴的观点研究拓扑群的重要性质。利用这一研究，我们证明了对于局部紧阿贝尔群$X$，下列条件是等价的：1）$X$是全不连通的，2）$\mathfrak{F}_0 (X)$是Schwartz群，3）$\mathfrak{F}_0 (X)$是紧的，4）$\mathfrak{F}_0(X)$具有Schur性质.因此，如果LCA群$X$有非零连通分支，则群$\mathfrak{F}_0(X)$是自反非Schwartz群，不具有Schur性质。我们还证明了，对于每一个紧连接可度量化阿贝尔群$X$的组$\mathfrak{F}_0 (X)$是monothetic的，推广了Rolewicz的结果$X=\mathbb{T}$。
英文摘要：
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 Š-space), then $\mathfrak{F}_0 (X)$ is an $(E)$-space (respectively, a strictly angelic space or a Š-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 $X=\mathbb{T}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
