Maximal vector topologies (Q429349)

Let \(V\) be the direct sum of \(k\) copies of \(\mathbb R\), where \(k\) is an arbitrary cardinal number. All considered topologies on \(V\) are topologies of vector \(\mathbb R\)-spaces where \(\mathbb R\) is endowed with the usual topology. There exist three canonical vector topologies on \(V\): (1) the box topology \(\tau_k\); (2) the maximal vector topology \(\mu_k\); (3) the maximal locally convex topology \(\nu_k\).NEWLINENEWLINEThe author studies properties of the topological vector spaces \((V,\tau_k)\), \((V,\mu_k)\), and \((V,\nu_k)\). The main results of the paper are:NEWLINENEWLINE(i) \(\nu_\omega=\tau_\omega\) and \(\nu_k\) is stronger than \(\tau_k\) for \(k>\omega\).NEWLINENEWLINE(ii) \((V,\tau_k), (V,\mu_k)\), and \((V,\nu_k)\) are complete.NEWLINENEWLINE(iii) For every infinite \(k\), the topological vector spaces \((V,\tau_k), (V,\mu_k)\), and \((V,\nu_k)\) are not sequential.NEWLINENEWLINE(iv) For every infinite cardinal number \(k\) there exists a family \(\mathcal F\) of locally convex topologies on \(V\) such that \(card \mathcal F=2^{2^k}\) and \(\tau_1\vee\tau_2=\nu_k\) for any distinct topologies \(\tau_1,\tau_2\) from \(\mathcal F\).