简单$sm$ -可因子分解（仿）拓扑群及其完备化
Simply $sm$-factorizable (para)topological groups and their completions
摘要：
我们称一个（对位）拓扑群为\emph{strongly submetrizable}，如果它允许一个较粗的可分可度量化（对位）拓扑群拓扑。利用连续实值函数给出了单$sm$ -可因子分解（仿）拓扑群的一个刻划.证明了（对位）拓扑群$G$是$sm$ -可单因子分解的当且仅当对于每个连续函数$f\colon G\to \mathbb{R}$，可以找到$G$到强次度量化（对位）拓扑群$H$和连续函数$g\colon H\to \mathbb{R}$上的连续同态$φ$，使得$f=g\circφ$ .这一刻画被应用于单PDMATHAMN-可因子分解拓扑群的完备化的研究。我们证明了等式$μ{G}=\varrho_ω{G}=\upsilon{G}$对每个Hausdorff单$sm$ -可因子分解拓扑群$G$成立.这一结果对Arhangel'skii和Tkachenko在2018年提出的一个问题给出了肯定的答案。此外，我们考虑了简单$sm$ -可分解的仿拓扑群的实紧化。证明了正则单$sm$ -可因子分解的仿拓扑群$G$的实紧化$\upsilon{G}$和Dieudonne完备化$μ{G}$是一致的，$\upsilon{G}$允许包含$G$的仿拓扑群的自然结构为稠密子群，而且$\upsilon{G}$也是单$sm$ -可因子分解的.改进或推广了[ \emph{Completions of paratopological groups, Monatsh. Math. \textbf{183} (2017), 699--721} ]中的一些结果.
英文摘要：
Let us call a (para)topological group \emph{strongly submetrizable} if it admits a coarser separable metrizable (para)topological group topology. We present a characterization of simply $sm$-factorizable (para)topo\-logical groups by means of continuous real-valued functions. We show that a (para)topo\-logical group $G$ is a simply $sm$-factorizable if and only if for each continuous function $f\colon G\to \mathbb{R}$, one can find a continuous homomorphism $φ$ of $G$ onto a strongly submetrizable (para)topological group $H$ and a continuous function $g\colon H\to \mathbb{R}$ such that $f=g\circφ$. This characterization is applied for the study of completions of simply $sm$-factorizable topological groups. We prove that the equalities $μ{G}=\varrho_ω{G}=\upsilon{G}$ hold for each Hausdorff simply $sm$-factorizable topological group $G$. This result gives a positive answer to a question posed by Arhangel'skii and Tkachenko in 2018. Also, we consider realcompactifications of simply $sm$-factorizable paratopological groups. It is proved, among other results, that the realcompactification, $\upsilon{G}$, and the Dieudonné completion, $μ{G}$, of a regular simply $sm$-factorizable paratopological group $G$ coincide and that $\upsilon{G}$ admits the natural structure of paratopological group containing $G$ as a dense subgroup and, furthermore, $\upsilon{G}$ is also simply $sm$-factorizable. Some results in [\emph{Completions of paratopological groups, Monatsh. Math. \textbf{183} (2017), 699--721}] are improved or generalized.
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。
