Axioms for Sequential Convergence

It is of general knowledge that those (ultra)filter convergence relations coming from a topology can be characterized by two natural axioms. However, the situation changes considerable when moving to sequential spaces. In case of unique limit points J. Kisyński [Kis60] obtained a result for sequential convergence similar to the one for ultrafilters, but the general case seems more difficult to deal with. Finally, the problem was solved by V. Koutnik [Kou85]. In this paper we present an alternative approach to this problem. Our goal is to find a characterization more related to the case of ultrafilter convergence. We extend then the result to characterize sequential convergence relations corresponding to Fréchet topologies, as well to those corresponding to pretopological spaces.