This is a well established concept in General Topology: "convergence structures"«convergence structures». The two references I would recommend are the first chapters of each one of the following books:
E. Binz, Continuous Convergence on $C(X).$$C(X)$. LNM Springer, 469.
R. Beattie and H.-P. Butzmann, Convergence Structures and Applications to Functional Analysis.
A quick overview: On a set $X$, for every $x\in X$ we define which filters converge to $x$, with the following restrictions: the ultrafilter of all supersets of $x$ must converge to $x$; any filter which contains a filter converging to $x$ must converge to $x$; the intersection of two filters converging to $x$ must converge to $x$ (contains" and intersection"«contains» and «intersection» to be understood in the usual set-theoretic sense).
Converging filters in a convergence subspace: a filter ${\mathcal F}$ converges to $x$ in the subspace iff the filter on the initial space generated by the filter basis ${\mathcal F}$ converges to $x$.
The sets which are present in every filter converging to $x$ are known as {\it neighborhoods}neighborhoods of $x$ with respect to the corresponding convergence structure $\Lambda$. Such sets are actually neighborhoods in the topological sense, for a topology on $X$ (called the topology associated with $\Lambda$). The definitions of a closed set as a set whose complementary is open, and as a set which coincides with its (filterwise) adherence, are equivalent. A set $K$ is said to be $\Lambda$-compact if every ultrafilter in $K$ converges with respect to the induced convergence structure on $K$. Filterwise continuous maps send compact sets onto compact sets.
