What is a Neighborhood? Which of these definitions is more commonly used, and in which contexts?
Fix a point $x\in (X, \tau)$. Then a neighborhood around a point $x$ is:


*

*a set $N\ni x$ and $N\in \tau$

*a set $N$ with $x\in \text{int}(N)$


If we are working in a space $(X, \tau)$ that is locally (path) connected:


*

*a set $N$ that is (path) connected and open

*a set $N$ that is simply (path) connected and open


Specifically, I am interested in the terminology that would be used in the study of PDEs such as in the book by Gilbarg and Trudinger.
Thanks!
 A: Just my feeling about it: a "neighborhood" of a point $x$ is intuitively a ball of some radius at $x$, and/but with potentially extra stuff farther away than we care about.
This is weird, but "it turns out to be useful". It is curiously hard to say why it does turn out to be mathematically useful to allow "balls of radius something" to be extrapolated to "balls of radius something, plus random stuff farther away", but, ... well, ... history? :)
Next up: path connected-ness? Well, usually this is of no moment...
In reference to a source such as G-and-T, it would matter whether you were refering to the _physical_space_ (on which some functions live), or/versus a topological vector space of functions on that space. Either way, I am inclined to think that "path-connectedness" is probably not the essential issue...
If I had to bet a dollar, it'd be toward having the questioner clarify whether their need for clarification is about the physical space on which their functions (to satisfying some PDE, and so on) live, or is about the topological vector space in which their functions live.
That is, I suspect there are "prior" issues...
Edit: Given further comments, etc, it becomes clear that some things aren't clear, namely, the contemporary defn of "neighborhood" does not include any overt assertion about path-connectedness. Nevertheless, in very nice spaces (locally Euclidean, for example), every neighborhood includes a path-connected neighborhood, indeed. Thus, in some contexts, writers want to build-in the path connectedness to the word "neighborhood" so that they don't say literally false things in discussion of path-connected neighborhoods, but don't have to repeat "path-connected" all the time.
Thus, it seems the the question is about usage, not about about path-connectedness itself, nor about the definition of "neighborhood"?
A: In topology, I have never seen "neighborhood" used to mean anything other than your second definition: a neighborhood of $x$ is a set containing $x$ in its interior. Your first definition, "open set containing $x$", is the definition of "open neighborhood of $x$". Your other definitions seem to have nothing to do with being a neighborhood at all.
A: I just ran across definition 2.3 (d), page 35 of Rudin's "Real and Complex Analysis", which defines a neighborhood as a synonym for "open set". It also comments that the term is not standardized. This is basically what everyone else explained to me here, but it may be helpful for others to see a reference (if someone had pointed me in this direction, my question would have been nipped in the bud).
