Discussion on question by ummg: Technical issue with empty metric space

@CyclotomicField including empty space as a metric space is not a choice: it is a fact that follows from the definition. Excluding empty space as ball is not a choice: it is a fact that follows from the definition. The only way to change this is to change definitions. But you gain nothing from that, those definitions are good as they are. You are incorrect claiming that the empty space is not useful as a metric space, it very much is. Unlike empty ball. I think I'm starting to repeat myself, so I will end this conversation now.