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On prebox module topologies. (English. Russian original) Zbl 1063.16051
Math. Notes 74, No. 1, 12-17 (2003); translation from Mat. Zametki 74, No. 1, 12-18 (2003).
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)].
MSC:
16W80 Topological and ordered rings and modules
46H05 General theory of topological algebras
54H13 Topological fields, rings, etc. (topological aspects)
54A10 Several topologies on one set (change of topology, comparison of topologies, lattices of topologies)
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