On prebox module topologies. (Q1889500)

Let \(k\) be a division ring with an absolute value \(|\cdot|\) and \(\tau\) the topology on \(k\) generated by \(|\cdot|\). For a vector \(k\)-space \(V\) and a fixed linear base \(X\) of \(V\) a \((k,\tau)\)-space topology \(\tau_b\) on \(V\) is defined as follows: A base at \(0\) for \(\tau_b\) consists of subsets \(U(f):=\sum_{x\in X}U_{f(x)}x\) where for each \(x\in X\) \(U_{f(x)}\) is a \(0\)-neighborhood of \((k,\tau)\) and \(f\colon X\to\omega\) is a mapping. The main result of the paper is: If \((k,\tau)\) is a complete topological division ring then \(V\) admits a \((k,\tau)\)-space topology \(\tau_1\) such that \(\tau_1<\tau_b\) and there is no \((k,\tau)\)-space topology \(\tau_2\) on \(V\) such that \(\tau_1<\tau_2<\tau_b\Leftrightarrow|X|\) is a measurable cardinal. Similar results were proved in the authors' papers [On premaximal topologies on vector spaces, Izv. Akad. Nauk Respub. Moldova Mat. 20, No. 1, 96-105 (1996) and Sib. Mat. Zh. 42, No. 3, 491-506 (2001; Zbl 1020.16033)].