Detecting topological groups which are (locally) homeomorphic to LF-spaces (Q386839)

A linear topological space \(X\) is called an LF-space if there exists a sequence \(\{X_n:n\in\omega\}\) of linear subspaces of \(X\) with the following properties:NEWLINE\begin{itemize}NEWLINE\item[\textbf{(lf1)}] \(X_i\subset X_{i+1}\) for every \(i\in\omega\);NEWLINE\item[\textbf{(lf2)}] every \(X_n\) is a locally convex completely metrizable space;NEWLINE\item[\textbf{(lf3)}] the topology on \(X\) is the strongest one that makes \(X\) a locally convex space for which all identity inclusions \(X_n\to X\) are continuous.NEWLINE\end{itemize}NEWLINEGiven a space \(X\), say that \(a\in X\) is a (strong) \(Z\)-point of \(X\) if for any open cover \(\mathcal U\) of the space \(X\), there exists a continuous map \(f:X\to X\) such that the set \(\{x,f(x)\}\) is contained in some element of \(\mathcal U\) for any \(x\in X\) and \(a\notin f(X)\) (\(a\notin \overline{f(X)}\) respectively).NEWLINENEWLINEIt is established that a topological group \(G\) is (locally) homeomorphic to an LF-space if there exists an increasing sequence \(\{G_n:n\in\omega\}\) of subgroups of \(G\) such thatNEWLINE\begin{itemize}NEWLINE\item[(1)] \(G=\bigcup_{n\in\omega}G_n\) and every \(G_n\) is (locally) homeomorphic to a Hilbert space;NEWLINE\item[(2)] if \(U_n\) is a neighborhood in \(G_n\) of the identity \(e\in G_n\), then \(\bigcup_{n=1}^\infty U_0U_1\ldots U_n\) is a neighborhood of \(e\) in \(G\).NEWLINE\item[(3)] for every \(n\in \mathbb N\), the quotient map \(G_n\to G_n/G_{n-1}\) is a locally trivial bundle;NEWLINE\item[(4)] for infinitely many numbers \(n\in \mathbb N\), every \(Z\)-point in the quotient space \(G_n/G_{n-1}\) is a strong \(Z\)-point.NEWLINE\end{itemize}