Raikov Remainders of Quotients by Raikov-Complete Almost Metrizable Normal Subgroups

Abstract

Let \(N\) be a closed normal subgroup of a topological group \(G\), and let \(\widehat q:\rho G\to\rho(G/N)\) extend the quotient homomorphism. We prove that \(r_\rho(G/N)=\widehat q\bigl(r_\rho(G)\bigr)\) 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 \(r_\rho(G)\) passes to \(r_\rho(G/N)\). In particular, the conclusion applies to closed locally compact normal subgroups.