I’ve read just the basics of some introductory analysis books and sometimes they show that we can characterize things like limits, continuity, compactness, etc. in terms of sequences.
I’ve heard that these sequential criteria hold for general metric spaces, but that in topology for example one encounters situations where sequences aren’t quite sufficient, or where it’s better to consider some other object.
My questions are:
- Is there some intuition for why the sequential criterion holds in things like Euclidean space or general metric spaces, but not in some other spaces?
- Does it simply have to do with the fact that we have a metric, and if so, why does the metric “induce” such sequential criteria (versus without a metric we may not)?
- Is the notion of distance/metric captured in some way by sequences because approaching some value is equivalent to a sequence approaching that value?
- Are there any ways by which we can determine whether a general space possesses these sequential criteria? They seem quite useful.