There is a short section in the book Locally Compact Groups by Markus Stroppel (Chapter B7) on the notion of a "Hausdorff Solvable Group", which he defines as a topological group with a descending chain of closed normal subgroups, but it ends rather abruptly. I've tried searching around for the Hausdorff Dervied Series but I can not find a reference other than in this book. Perhaps this goes by a different name in other texts, but assuming not:

If we equip the automorphism group of a field extension with the compact-open topology (by assumption of material present in the same book, chapter C9), then this shares separation properties of the field (9.2). What is the significance of an automorphism group being Hausdorff Solvable, or being solvable but not Hausdorff Solvable? Can we construct some examples?