Yes of course, I didn't mean that we should stick to metrizable spaces. Though it's intuitive for humans and its limits are quite broad, if I may say :) Eg. w

Yes of course, I didn't mean that we should stick to metrizable spaces. Though it's intuitive for humans and its limits are quite broad, if I may say :) Eg. we usually don't start defining the derivative with distributions. I didn't mention geometry neither btw though it's foundational too imo.