Let $R$ be a ``topological PID'': a topological ring which is an integral domain in which every principal ideal is closed and every closed ideal is principal.

Is $R$ ``topologically Noetherian'': there is no strictly increasing sequence of closed ideals?

If not, is there a locally compact counterexample?

Romain Giquaud has given a counterexample to the general form of the question. The bounty is for a solution for locally compact, metrizable rings. (I suspect the answer may be positive with this restriction.)

Let $R$ be a ``topological PID'': a topological ring which is an integral domain in which every principal ideal is closed and every closed ideal is principal.

Is $R$ ``topologically Noetherian'': there is no strictly increasing sequence of closed ideals?

If not, is there a locally compact counterexample?