Bert Mendelson's "Introduction to Topology" (3ed, 1975, Dover pubs) spends a section in both his chapter on metric spaces and that on general topological spaces (chapters 2 and 3 respectively) hammering out the axioms for a neighborhood space.
A neighborhood space is a set $S$ such that, for each $x \in S$, there exists a set of subsets $\mathcal N_x$ of $S$ satisfying the following conditions:
$\text N 1$: There exists at least one element in $\mathcal N_x$
$\text N 2$: Each element of $\mathcal N_x$ contains $x$
$\text N 3$: Each superset of $N \in \mathcal N_x$ is also in $\mathcal N_x$
$\text N 4$: The intersection of $2$ elements of $\mathcal N_x$ is also in $\mathcal N_x$
$\text N 5$: There exists $N' \subseteq N \in \mathcal N_x$ which is in $\mathcal N_y$ for each $y \in N'$
He goes on to establish that a neighborhood space is exactly the same as a topological space: the neighborhood space induced by the topological space induced by a neighborhood space is that neighborhood space, and so on.
So the neighborhood space axioms define exactly the same space as the well-known open set axioms of topology.
Proving that a collection of subsets of a set adhere to the neighborhood space axioms is fiddly and tedious, and conceptually non-intuitive. Particularly axiom $N5$, which is horrible.
The question is: is there a good reason for a "neighborhood space" to be defined? Does it have any particular uses that make it essential or worthwhile to study in detail? Or is it just an interesting backwater that Mendelson (and also Willard, it seems, by taking a glance at the Wikipedia page on "Neighborhood system") defines for the fun of it?
Hence, how much effort would one be expected to exert in an attempt to master fluency in handling such constructs?
(No doubt there may be some difficult-to-analyse spaces which are analysed more easily using neighborhood axioms rather than open set axioms -- but are they sufficient bang for the buck?)