On countably complete topological groups (Q6592031)

Given a topological space \(X\), let \(\tau(X)\) be its topology and \(\tau^*(X) = \tau(X) \setminus\{\emptyset\}\). The space \(X\) is called \textit{countably complete} if there exists a sequence \(\{\mathcal U_n: n\in\omega\}\) of open covers of \(X\) such that \(\bigcap_{n\in\omega}\overline O_n \neq\emptyset\) whenever \(\{O_n: n\in\omega\} \subset \tau^*(X)\) is a decreasing sequence such that for every \(n\in\omega\), there is \(k\in\omega\) with \(O_k\subset U\) for some \(U\in \mathcal U_n\). A subspace \(Y\subset X\) is \textit{relatively pseudocompact} if for any infinite family \(\mathcal U\subset \tau(X)\) such that \(U\cap Y\neq\emptyset\) for each \(U\in \mathcal U\), there is \(y\in Y\) which is a cluster point of \(\mathcal U\), i.e., every set \(V\in \tau(X)\) with \(x\in V\), meets infinitely many elements of \(\mathcal U\).\N\NIt is said that \(x\in X\) is a \textit{pseudocompactness point of \(X\)} if there exists a sequence \(\{U_n: n\in\omega\}\) of open neighborhoods of \(x\) such that a sequence \(\{V_n: n\in \omega\}\subset \tau^*(X)\) has a cluster point in \(X\) given that \(V_n\subset U_n\) for every \(n\in\omega\). The space \(X\) is \textit{pointwise pseudocompact} if every \(x\in X\) is a pseudocompactness point of \(X\). A set \(A\subset X\) is \textit{\(C\)-embedded } in \(X\) if each continuous function \(f:A\to \mathbb R\) continuously extends over \(X\), i.e., there is a continuous function \(g:X\to \mathbb R\) such that \(g|A=f\).\N\NThe authors prove, among other things, that a topological group \(G\) is countably complete if and only if \(G\) contains a relatively pseudocompact subgroup \(H\) such that the quotient space \(G/H\) is completely metrizable and the inverse images under the quotient map \(\pi:G\to G/H\) keep their relative pseudocompactness, i.e., \(\pi^{-1}(Y)\) is relatively pseudocompact in \(G\) for any relatively pseudocompact subspace \(Y\) of the space \(G/H\). Another result of the paper reads that for any pointwise pseudocompact group \(G\), if \(H\) is a countably complete subgroup of \(G\), then \(H\) is \(C\)-embedded in \(G\).