From what I understand, the point of first countability is to ensure sequences capture all topological information. It's not too difficult to prove that a set function from a first countable space to any space is continuous if and only if it's sequentially continuous.
My question is: does this equivalence characterize first countable spaces? I'm guessing it does but have no idea how to prove so.
It does not: it characterizes sequential spaces, as is noted in this answer. This isn’t too hard to prove; once you get the definition of sequential space clear in your mind, you might try to prove it.
Sequential spaces are precisely those in which the convergent sequences completely determine the topology, so this is not really surprising. All first countable spaces are sequential, but there are many sequential spaces that are not first countable. Dan Ma’s Topology Blog has a series of informative posts on sequential spaces, starting here.
