Reflecting some properties of topological groups (Q2400889)

A topological group \(G\) is called \textit{\(\omega\)-narrow,} if for any neighborhood \(U\) of the identity of \(G\), there exists a countable set \(A\subset G\) such that \(AU=G\). Given an \(\omega\)-narrow topological group \(G\) and a cardinal function \(\varphi\in \{\)weight, cellularity, character, pseudocharacter, tightness, \(o\)-tightness\(\}\), it is established that, for any regular infinite cardinal \(\kappa \leq\varphi(G)\), there exists a continuous homomorphism of \(G\) onto a topological group \(H\) such that \(\varphi(H)=w(H)=\kappa\). It is also shown that an \(\omega\)-narrow topological group \(G\) is pseudocompact (precompact) if and only if all continuous second countable homomorphic images of \(G\) are pseudocompact or precompact respectively.