For metric spaces, knowledge of the convergence of sequences determines the topology completely. A set is closed in the metric topology if and only if it is closed under the limit of convergent sequences operation. Put another way, a map between metric spaces is continuous if and only if it preserves limits of sequences.
For general topological spaces, this result no longer holds. Spaces for which it does hold are called sequential spaces. Spaces are usually not sequential because they are not first countable. In such spaces, there are "not enough" numbers in the countable index set to extend into these very large spaces and reach every limit point. For example, if $\omega_1$ is the first uncountable ordinal, then $\omega_1+1$ is a topological space under the order topology, and $\omega_1$ is a limit point, but no sequence of ordinals reaches $\omega_1$. The topology of this space is not characterized by limits of sequences.
This defect is usually fixed by replacing the notion of a sequence with the more general notion of a net, which is like a sequence but indexed by an arbitrary directed set, instead of the natural numbers. Since the net's index set is arbitrary, it can be of any cardinality and therefore nets can reach those faraway limit points in non first countable spaces, and the result is restored: a function between topological spaces is continuous iff it preserves limits of nets. A set is closed iff it is closed under limit of nets.
But if the problem is that the cardinality of the space is too high for sequences to do the job, the most conservative generalization that suggests itself to me is to just increase the cardinality of the index set; the transfinite sequence, indexed by an arbitrary ordinal. So my question is: is this strong enough? Is it just a matter of convenience? Nets require less machinery, and are natural because e.g. the neighborhoods of a point constitute a net, whereas transfinites require a detour into unrelated areas of set theory. But nets are not totally ordered, and there may be times it would be nice to have a totally ordered index set for our sequence-like object.
So can we say that a set in an arbitrary topological space is closed iff it's closed under taking limits of arbitrary convergent transfinite sequences? Is there a space with a set which is closed under limit of transfinite sequence, but not limit of net?