Characterization of a topology via its convergent sequences

Hello everyone, heres a question that has been troublying me for quite a while, and also one that I hope some of you folks can answer! Here it goes:

I found out recently that a topology can be characterized both by its convergent filters and/or its convergent nets. From this point one is led to consider abstract "filter convergence spaces", specifying certain axioms that a filter or net convergence relation must satisfy as a minimun requisite to be "convergence". I learned also that you can impose adittional axioms on the convergence relation so that there exists a topology having precisely those convergence filters/nets, so that the abstract convergence relation is "topological".

My question is: can you do the same thing but considering only sequences? That is: can you fully characterize the topology of, say, a first countable space by prescribing a notion of convergence of sequences, defining the topology in some way from the convergence relation, and then proving that a sequence converges according to the convergence relation if and only if it converges with respect to the topology just defined?