14.5 (1999)

Let $G$ be an infinite group admitting non-discrete Hausdorff group topologies, and $\mathscr{L}(G)$ the lattice of all group topologies on $G$.

a) Is it true that for any natural number $k$ there exists a non-refinable chain $\tau_0 < \tau_1 < \dots < \tau_k$ of length $k$ of Hausdorff topologies in $\mathscr{L}(G)$? (For countable nilpotent groups this is true (A. G. Topale, Deposited in VINITI, 25.12.98, no. 3849–V 98 (Russian)).)

b) Let $k, m, n$ be natural numbers and let $G$ be a nilpotent group of class $k$. Suppose that $\tau_0 < \tau_1 < \dots < \tau_m$ and $\tau'_0 < \tau'_1 < \dots < \tau'_n$ are non-refinable chains of Hausdorff topologies in $\mathscr{L}(G)$ such that $\tau_0 = \tau'_0$ and $\tau_m = \tau'_n$. Is it true that $m \leqslant n \cdot k$ and this inequality is best-possible? (This is true if $k = 1$, since for $G$ abelian the lattice $\mathscr{L}(G)$ is modular.)

c) Is it true that there exists a countable $G$ such that in the lattice $\mathscr{L}(G)$ there are a finite non-refinable chain $\tau_0 < \tau_1 < \dots < \tau_k$ of Hausdorff topologies and an infinite chain $\{\tau'_\gamma \mid \gamma \in \Gamma\}$ of topologies such that $\tau_0 < \tau'_\gamma < \tau_k$ for any $\gamma \in \Gamma$?

d) Let $G$ be an abelian group, $k$ a natural number. Let $\mathcal{A}_k$ be the set of all those Hausdorff group topologies on $G$ that, for every topology $\tau \in \mathcal{A}_k$, any non-refinable chain of topologies starting from $\tau$ and terminating at the discrete topology has length $k$. Is it true that $\mathcal{A}_k \cap \{\tau'_\gamma \mid \gamma \in \Gamma\} \neq \varnothing$ for any infinite non-refinable chain $\{\tau'_\gamma \mid \gamma \in \Gamma\}$ of Hausdorff topologies containing the discrete topology? (This is true for $k = 1$.)