There is a literature on "convergence spaces" of various kinds. I read some of that in the 70s, but I do not remember a lot of the detail. There is something called "pre-topological space" or "closure space". And there is "pseudo-topological space". Each of them can be defined in terms of convergence of filters. Or in terms of convergence of nets. Or in terms of neighborhood systems. Or in terms of a closure operation.. One of these is associated with Choquet. There is a big topology text Topological Spaces by Gál Cech that takes the closure space as the fundamental notion.
Added November 16.
From my old paper "Three Crypotisomorphism Theorems" in Studies in Foundations and Combinarorics, Advances in Mathematics Studies vol. 1, 1978, pp. 49--60
[Pretopology, closure space, mehrstufige Topologie, pré-adhérence]
Axioms in terms of a closure operation $\eta$ from sets to sets:
(b1) $A \subseteq \eta(A)$
(b2) $\eta(\emptyset) = \emptyset$ and $\eta(A \cup B) = \eta(A) \cup \eta(B)$.
Of course to specify a topology, we have to add a third axiom $\eta(\eta(A)) = \eta(A)$. Without the third axiom, we get the more general pretopology.