论文标题
预校准阿贝尔群体的密度和权重
Densities and Weights of Quotients of Precompact Abelian Groups
论文作者
论文摘要
著名的Banach-Mazur问题的拓扑组版本询问每个无限拓扑组都有一个非平凡的可分离商组。众所周知，紧凑型组具有无限可分离的可分离的商组。但是，作为紧凑型组的密集亚组，预反感组可能不承认不可动摇的商组，因此也没有非平凡的可分离商组。在本文中，我们研究了最少的红衣主教$ \ mathfrak {m} $（resp。$ \ m rathfrak {n} $），使每个无限的预发Abelian Group都接纳了一个具有密度符号$ \ leq \ leq \ Mathfrak {m Mathfrak {m Mathfrak {m} $的商集团（resp。结果表明，如果$ 2^{<\ Mathfrak {C}} = \ Mathfrak {C} $，则$ \ Mathfrak {M} = \ Mathfrak {C} $ {C} $和$ \ Mathfrak {n} = 2^\ Mathfrak {C} $。 一个更普遍的问题是描述所有可能的无限符号的$ QW（G）$ QW（g）$，该$ QW（g）$ g $ abelian $ g $。我们证明，对于间隔$ [ω，\ mathfrak {c}] $的每个子集$ e $，都存在一个预先校准的Abelian Group $ g $，带有$ qw（g）= e $。如果$ω\在e $中，则可以选择$ g $作为假发。 在附录中，我们举例说明非全面断开的本地紧凑型组可能会承认没有可分离的商组。这回答了\ cite {lmt}中提出的一个开放问题。
The topological group version of the celebrated Banach-Mazur problem asks wether every infinite topological group has a non-trivial separable quotient group. It is known that compact groups have infinite separable metrizable quotient groups. However, as dense subgroups of compact groups, precompact groups may admit no non-trivial metrizable quotient groups, so also no non-trivial separable quotient groups. In this paper, we study the least cardinal $\mathfrak{m}$ (resp. $\mathfrak{n}$) such that every infinite precompact abelian group admits a quotient group with density character $\leq \mathfrak{m}$ (resp. with weight $\leq \mathfrak{n}$). It is shown that if $2^{<\mathfrak{c}}=\mathfrak{c}$, then $\mathfrak{m}=\mathfrak{c}$ and $\mathfrak{n}=2^\mathfrak{c}$. A more general problem is to describe the set $QW(G)$ of all possible weights of infinite proper quotient groups of a precompact abelian group $G$. We prove that for every subset $E$ of the interval $[ω, \mathfrak{c}]$, there exists a precompact abelian group $G$ with $QW(G)=E$. If $ω\in E$, then $G$ can be chosen to be pseudocompact. In an appendix, we give an example to show that a non-totally disconnected locally compact group may admit no separable quotient groups. This answers an open problem posed in \cite{LMT}.
