Has convergence of sequences been axiomatized in this or a similar manner?

Question

Let $S$ be a nonempty set, and consider the set $S^\mathbb{N}$ of all infinite sequences with elements in $S$. I shall call that set $Seq(S)$. Suppose $f$ is a partial function from $Seq(S)$ to $S$, not a total function, but a partial function. $f$ is said to be a convergence function if the following criteria are satisfied:

1. $f(c,c,c,c,...)=c$ for all $c\in S$.
2. If $f(s_n)=x$, then for any sequence $t_n$ which is obtained by deleting, adding, and/or rearranging finitely many elements of $s_n$, we also have $f(t_n)=x$.
3. If $f(s_n)=x$, then $f(u_n)=x$ for every subsequence $u_n$ of $s_n$.

My question is, has this notion of convergence of sequences been axiomatized, either like this or in a similar manner, in the mathematical literature? Perhaps they use a different set of axioms, or more axioms than mine. I would be very interested to read such a text that talks about this.