论文标题
在封闭的前组中
On closed subgroups of precompact groups
论文作者
论文摘要
这是W.〜W。Comfort和K.〜A。Ross的定理，如果$ g $是紧凑型Abelian集团的一个子组，而$ S $表示这些连续的同型从$ g $到一维圆环，那么$ G $的拓扑是$ g $的最初拓扑。 {假设$ h $是$ g $的子组。我们研究$ s $的选择如何影响$ g $中$ h $的拓扑放置和属性。除其他结果外，我们在解决以下特定问题的解决方案方面取得了重大进展：$ g $允许多少个完全有限的群体拓扑，使得$ h $是一个封闭的（密集）亚组？如果$ c_s $表示$ g $的所有子组的poset，则为$ s $ cluct，按包含订购，$ c_s $是否具有最大的（分别为最小的）元素？我们说，如果关闭其所有子组，则一个完全有限的（拓扑，分别）组是一个\ textit {sc-group}（\ textit {topologicyssip simple}，resp。）。任意的Abelian集团$ g $会接受吗？
It is a Theorem of W.~ W. Comfort and K.~ A. Ross that if $G$ is a subgroup of a compact Abelian group, and $S$ denotes those continuous homomorphisms from $G$ to the one-dimensional torus, then the topology on $G$ is the initial topology given by $S$. {Assume that $H$ is a subgroup of $G$. We study how} the choice of $S$ affects the topological placement and properties of $H$ in $G$. Among other results, we have {made significant} progress toward the solution of the following specific questions: How many totally bounded group topologies does $G$ admit such that $H$ is a closed (dense) subgroup? If $C_S$ denotes the poset of all subgroups of $G$ that are $S$-closed, ordered by inclusion, does $C_S$ has a greatest (resp. smallest) element? We say that a totally bounded (topological, resp.) group is an \textit{SC-group} (\textit{topologically simple}, resp.) if all its subgroups are closed (if $G$ and $\{e\}$ are its only possible closed normal subgroups, resp.) {In addition, we investigate the following questions.} How many SC-(topologically simple totally bounded, resp.) group topologies does an arbitrary Abelian group $G$ admit?
