A HIERARCHY ON NON-ARCHIMEDEAN POLISH GROUPS ADMITTING A COMPATIBLE COMPLETE LEFT-INVARIANT METRIC

In this article, we introduce a hierarchy on the class of non-archimedean Polish groups that admit a compatible complete left-invariant metric. We denote this hierarchy by alpha-CLI and L-alpha-CLI where alpha is a countable ordinal. We establish three results: (1) G is 0-CLI iff G = {1(G)}; (2) G is 1-CLI iff G admits a compatible complete two-sided invariant metric; and (3) G is L-alpha-CLI iff G is locally alpha-CLI, i.e., G contains an open subgroup that is alpha-CLI. Subsequently, we show this hierarchy is proper by constructing non-archimedean CLI Polish groups G(alpha) and H-alpha for alpha < omega(1), such that: (1) H-alpha is alpha-CLI but not L-beta-CLI for beta < alpha; and (2) G(alpha) is (alpha+1)-CLI but not L-alpha-CLI.