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One problem with epsilon-delta definitions is that they don't scale. In many topological spaces, there's no concept of distance and so epsilon-delta stuff has t
by ifbutthen 5y ago
One problem with epsilon-delta definitions is that they don't scale. In many topological spaces, there's no concept of distance and so epsilon-delta stuff has to be replaced with more general open sets. About the same reason why sequences have to be replaced with nets and filters.
- ronald_raygun 5y agoyeah there are like 10 million ways of defining cont. or limits in R^N that turn out to be the same
- ifbutthen 5y agoR^n is Euclidean and so epsilon-delta defns work just fine. I am talking about topologies like cofinite topology or cocountable topology where the picture doesn't look like a grid to which you can apply distance related notions of epsilon- room here and delta-long there.
- rq1 5y agoYes 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.