Raikov完备的几乎可度量化正规子群所得商群的Raikov余项
Raikov Remainders of Quotients by Raikov-Complete Almost Metrizable Normal Subgroups
AI总结：
本文研究拓扑群商群的Raikov余项性质，刻画了商同态延拓下余项等式成立的充要条件，证明Raikov完备几乎可度量化闭正规子群满足该条件，得出伪紧性可传递的结论。
AI中文摘要：
设\textit{N}是拓扑群\textit{G}的闭正规子群，\u0302\textit{q}:ρ\textit{G}→ρ(\textit{G}/\textit{N})是商同态的延拓。我们证明：\textit{r}_ρ(\textit{G}/\textit{N})=\u0302\textit{q}(\textit{r}_ρ(\textit{G}))成立当且仅当\u0302\textit{q}是满射且\u0302\textit{q}⁻¹(\textit{G}/\textit{N})=\textit{G}，且两个条件互不蕴含。当\textit{N}是Raikov完备且几乎可度量化时，两个条件均满足。由此，\textit{r}_ρ(\textit{G})的伪紧性可传递给\textit{r}_ρ(\textit{G}/\textit{N})。该结论特别适用于闭局部紧正规子群。
英文摘要：
Let \(N\) be a closed normal subgroup of a topological group \(G\), and let \(\widehat q:ρG\toρ(G/N)\) extend the quotient homomorphism. We prove that \(r_ρ(G/N)=\widehat q\bigl(r_ρ(G)\bigr)\) holds if and only if \(\widehat q\) is onto and \(\widehat q^{-1}(G/N)=G\), and we show that neither condition implies the other. Both conditions hold whenever \(N\) is Raikov complete and almost metrizable. Consequently, pseudocompactness of \(r_ρ(G)\) passes to \(r_ρ(G/N)\). In particular, the conclusion applies to closed locally compact normal subgroups.
