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.
As follows:
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?)
Best Answer
You seem to have already made up your mind that this isn't worth your time, so it's hard to know what can be provided as an answer. But I would say that this is worth your time if for no other reason than you are clearly struggling to translate between two axiomatic systems that describe the same thing. That is a fundamental mathematical skill and anyone with a Ph.D. will tell you that they needed it, in some form or another, to get where they are.
I personally needed this definition, and at least a dozen other similar ones, for my thesis work. Let me try to explain why without getting into too much detail. The key point is this: Different sets of axioms generalize differently, even if the axioms themselves are equivalent. You are already starting to see this when you call axiom N5 "horrible"; indeed, in this way of viewing topology, filters are natural, but open sets are not. Obviously, in the most popular definition of a topological space, open sets are taken as fundamental.
Before 1990, rigid analytic geometry (which is very far from being a backwater) was largely studied by means of Grothendieck topologies, which are more or less a big categorical generalization of filters or neighborhood spaces. This is a bit strange, as there were actual topological spaces. But these spaces didn't behave in intuitive ways, for example they failed to be locally compact, an important technical property that went through just fine in the more abstract setting.
But Berkovich demonstrated that we can add some points (well, a lot of points) to get a really nice topological space, one that is locally compact and Hausdorff. Why was this possible, and why did nobody notice this before?
The key lies in something like axiom N5. You see, traditional non-archimedean spaces have, in a sense, too many open sets. The whole point of using Grothendieck topologies was to ensure that not every open set containing a point was a neighborhood of that point, by forcing it to contain some "much smaller" neibghborhood (in the literature this is sometimes called overconvergence).
Somehow, mathematicians knew that these open sets were not "open enough", so they had to be somehow excluded. Now we sort of know why: a (nice) classical non-archimedean space $X$ is a dense subspace of a locally compact Hausdorff space $\hat{X}$ (whose points may be difficult to get a handle on concretely). In particular, some open sets/coverings in $X$ are special, namely the preimages of open sets/coverings in $\hat{X}$. In fact, this means that the space $\hat{X}$ was always somehow implicit in a neighborhood system on $X$! (i.e. $\hat{X}$ consists of the points of some topos, blah blah blah)
For me, the moral of this story is that doing mathematics is not just about learning the most popular axioms and drawing conclusions from them. It's about understanding the interplay between different definitions. New research is often about forging new connections, and it's held back often precisely because some area is considered "backwater" and so on. Mathematics can go in or out of style, but it rarely becomes truly obsolete. Don't just chase style, take the time to understand the various perspectives (even the silly ones) and I can promise you it will pay off.