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On coverings in the lattice of linear topologies. (English)

Authors: Arnautov Vladimir, Filippov K. M.

Abstract

Let $M$ be a module over a discrete skew-field, $\goth L(M)$ be the lattice of all linear topologies on $M$. For an arbitrary $\tau_0\in\goth L(M)$ and an arbitrary coatom $\tau_*\in \goth L(M)$, the relation $\inf\{\tau_0,\tau_*\}\prec\tau_0$ holds, i.e. $\tau_0$ covers $\inf\{\tau_0,\tau_*\}$. Let $\tau_0\in\goth L(M)$. The question is whether for every $\tau_1\in\goth L(M)$, $\tau_1\prec\tau_0$ there exists a coatom $\tau_*\in\goth L(M)$ such that $\tau_1=\inf\{\tau_0,\tau_*\}$. The answer is proved to be negative for every case when topological module $(M,\tau_0)$ is Hausdorff and non-complete, and positive for every case when it is Hausdorff, complete and metrizable.

Institute of Mathematics and Computer Science