发表机构
School of Mathematics and Statistics, Hanjiang Normal University（汉江师范学院数学与统计学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明对任意序数与阶段集合，可构造伪紧布尔群及其闭正规子群塔，使 Raikov 余集的非空层精确出现在指定阶段，并给出纤维同胚分解。
AI 中文摘要
设 $\theta$ 为一个序数，并设 $(N_\alpha)_{\alpha\leq\theta}$ 为 Hausdorff 拓扑群 $G$ 的闭正规子群的一个连续递增塔，其中 $N_0=\{e\}$。记 $\rho G$ 为 $G$ 的 Raikov 完备化，并令 $K_\alpha=\overline{N_\alpha}^{\rho G}$ 且 $H_\alpha=GK_\alpha$。群 $H_\alpha$ 的相继差与极限差标记了 $\rho G$ 中的点成为旧商点的最初阶段。对于每个非零序数 $\theta$ 和每个集合 $S\subseteq(0,\theta]$，存在一个 Hausdorff 伪紧布尔群，具有闭正规子群的连续塔，使得塔的每个成员都是伪紧的，且非空层恰好出现在 $S$ 中的阶段。终端商可以同时被选择为同构于 $\mathbb Z/2\mathbb Z$，并且对于每个 $\delta<\theta$，相继因子 $N_{\delta+1}/N_\delta$ 是紧的当且仅当 $\delta+1\notin S$。因此，即使每个相继因子都是紧的，一个塔也可以在极限阶段具有单个非空层。对于 $G$ 的闭正规子群 $N$，令 $\widehat q:\rho G\to\rho(G/N)$ 扩展商同态，并令 $K=\overline N^{\rho G}$。在 $gN\in G/N$ 上，$\rho G\setminus G$ 中的纤维是 $g(K\setminus N)$，同胚于 $\rho N\setminus N$，而对于 $y\in\widehat q(\rho G)\setminus(G/N)$，整个 $\widehat q$-纤维位于 $\rho G\setminus G$ 中，并且同胚于 $\rho N$。对于预紧核，这给出了一个典范分解，嵌套商给出了超限过滤背后的两阶段恒等式。进一步的结果涉及伪紧 Raikov 余集，并包括可数阿贝尔例子。
英文摘要
Let $θ$ be an ordinal and let $(N_α)_{α\leqθ}$ be a continuous increasing tower of closed normal subgroups of a Hausdorff topological group $G$, with $N_0=\{e\}$. Write $ρG$ for the Raikov completion of $G$, and put $K_α=\overline{N_α}^{ρG}$ and $H_α=GK_α$. The successive and limit differences of the groups $H_α$ mark the first stages at which points of $ρG$ become old quotient points. For every nonzero ordinal $θ$ and every set $S\subseteq(0,θ]$, there is a Hausdorff pseudocompact Boolean group with a continuous tower of closed normal subgroups such that every member of the tower is pseudocompact and the nonempty strata occur exactly at the stages in $S$. The terminal quotient can simultaneously be chosen isomorphic to $\mathbb Z/2\mathbb Z$, and for every $δ<θ$ the successive factor $N_{δ+1}/N_δ$ is compact exactly when $δ+1\notin S$. Thus a tower may have a single nonempty stratum at a limit stage even when every successive factor is compact. For a closed normal subgroup $N$ of $G$, let $\widehat q:ρG\toρ(G/N)$ extend the quotient homomorphism and put $K=\overline N^{ρG}$. Over $gN\in G/N$, the fiber in $ρG\setminus G$ is $g(K\setminus N)$, homeomorphic to $ρN\setminus N$, while for $y\in\widehat q(ρG)\setminus(G/N)$ the entire $\widehat q$-fiber lies in $ρG\setminus G$ and is homeomorphic to $ρN$. For precompact kernels this gives a canonical decomposition, and nested quotients give the two-stage identity underlying the transfinite filtration. Further results concern pseudocompact Raikov remainders and include countable Abelian examples.
Comments24 pages
