I’m not really confused about what you’re saying here exactly, and since the original post is deleted, I can’t really even see what was originally said, but I was confused about this:
(like the real number that that sequence converges to given the standard topology of the space of real numbers).
Why make mention of the standard topology here exactly? It’s not exactly clear to me why this has anything to do with what you two are discussing.
Thanks for the clarification! In my mind, I sort of just think “metric first” so the topology induced by that metric is always just assumed, but that’s because I don’t ever work with non-metrizable spaces.