Topological groups with a \(bc\)-base (Q465844)

This article deals with a question posed by L. G. Zambakhidze: \textit{Is every non-zero-dimensional topological group with a \({bc}\)-base locally compact?}NEWLINENEWLINEThe authors show that the small inductive dimension \textit{ind} of any non-locally compact group with a \({bc}\)-base does not exceed 1. They also prove that a \(\sigma\)-compact non-locally compact topological group with a \({bc}\)-base is zero-dimensional. Furthermore, they add to more important results:NEWLINENEWLINE1) If the free topological group \(F(X)\) of a Tychonoff space \(X\) has a \({bc}\)-base, then \(\text{ind} (X)\leq 0\), and 2) a topological group \(G\) has a \({bc}\)-base if and only if \(G\) can be compactified by a zero-dimensional remainder.