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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!

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Personally I think it is a matter of technicality in proofs for the neighborhoods definition, and actually never really matters because the whole idea of a neighborhood is that you can know what happens when you are epsilon-close to $x$. What's "far away" from $x$ usually isn't important, so the first two definitions are different yes, but the difference is irrelevant to most problems. –  Patrick Da Silva Jul 20 '12 at 22:50
    
And have you ever run across the definitions involving connectivity? Which contexts are you most familiar with, just in general topology or also in applications? –  pre-kidney Jul 20 '12 at 22:52
    
-1 Voting to close as not a real question. –  user31373 Jul 21 '12 at 1:51
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I think that the question is quite suitable, but perhaps improperly tagged, as "neighborhood" is a loose term used frequently. –  Arkamis Jul 21 '12 at 2:30
    
@Leonid : The question is the first line. Saying that OP's question is not a real question is just telling him that he's asking something stupid. Think a little before downvoting something like that. –  Patrick Da Silva Jul 21 '12 at 4:38

3 Answers 3

up vote 1 down vote accepted

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.

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I believe I have seen a few authors using the term neighborhood for what you would call open neighborhood. My copy of Munkres says on p.96 (following Theorem 17.5, which describes closure of a set): Mathematicians often use some special terminology here. They shorten the statement "$U$ is an open set containing $x$" to the phrase "$U$ is a neighborhood of $x$." But I agree that the usage you describe is more usual. –  Martin Sleziak Jul 21 '12 at 9:51

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"?

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Whoa! Someone down-voted! High-larious... I suppose there is "evil", if not actual evil, in the world. Pity. –  paul garrett Jul 21 '12 at 1:56
    
Absorbed the downvote with my own upvote. But the only reason I did this is because downvoter didn't explain himself, which I hate ; I actually think your answer's not that good ; you didn't say which definition was more used that the other one and in which context ; that was OP's question. Perhaps it was too big for a comment (such as mine), but then in that case you can just mention "This was too big for a comment, so I'll answer it instead..." –  Patrick Da Silva Jul 21 '12 at 4:39

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).

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