At the beginning the concept of limit is introduced for sequences thanks to the fact that the space in which we operate is a metric space. Then we introduce topology and we can define limits using neighbourhoods, and we see that any metric space is a topological space with a countable base of neighbourhoods (i.e. is a first-countable space). For a topological space that is not first-countable we can use nets and filters that generalize the concept of sequence, but we are always in a topological space. There is some way to define the concept of limit of a sequence in a set where a topology is not defined?
Look up convergence space (e.g. here). This axiomatises the convergence of filters (or nets).
There is even a notion of convergence space just for sequences (instead of nets or filters). Fréchet defined these first, I think. Here the notion of convergence of a sequence is axiomatised. There are then special convergence spaces where the convergence is induced by a topology.
In a convergence space we have a few basic axioms:
- the constant sequence converges to the constant.
- if we modify a sequence by finitely many terms, and one sequence converges to some limit, the other one does too.
- if a sequence converges to a limit, so does every subsequence of that sequence.
(I believe these are all the original ones.)
In order to be induced by a topology, we need the axiom that : if a sequence $(x_n)$ is such that every subsequence of it has itself a subsequence that converges to $x$, then the original sequence $(x_n)$ converges to $x$. Such a topology then need not be unique (as topologies need not be uniquely determined by the convergence of its sequences).
This nice answer assumes the last axiom (as L3) as well, and shows how to define a topology from it, and gives some references. There is another axiom to make it even more well-behaved (his axiom L4).
For nets we have similar axioms, and for filters too. And there we have exact characterisations for when the convergence is topological, as these convergence notions do chacterise topologies (again, one needs some extra axioms as for sequences). Kelly's book on Topology has some theory on this, IIRC. The sequential convergence spaces theory is somewhat obscure and mostly historically relevant (when they were not in agreement about what the right generalisation for a topology was).
Indeed, it is classical that convergence almost everywhere on Euclidean Lebesgue measure spaces has the property that it defines a convergence space that cannot be induced by a topology (as it does not satisfy the special subsequence of subsequence axiom).