writes "I consider this as evidence that you should not ignore the topology."
We should never ignore the topology of anything. 🙂
Stone said "always topologize", and he was right.
Almost right. Things have an intrinsic topology, which is not our choice.
This becomes aparent when we work constructively, where the intrinsic topologies we get are the topologies classical people claim to impose. But there was no choice, really. The topology is always intrinsic. It is not something we choose.