Talk:Neighbourhood (mathematics)

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Issues with recent additions to this article (request for comment)[edit]

I have mixed feelings about the recent changes to this article. It looks to me that the text is unnecessarily complicated, and the part contained in the ==Definition== section is no definition at all, rather some generalities, which I am having trouble following. Then, the way the article is now, one starts with the euclidean space, only to go to the topological space, then followed by a metric space, then back to the real numbers.

Is there any way to integrate the recently inserted material? Oleg Alexandrov 02:44, 28 July 2005 (UTC)

Punctured Neighbourhood[edit]

I would like to add Puncture Neighbourhood, but I just fail. It is the same as a normal neighbourhood, only it does not actually contain p.

"wiggle" or "move"[edit]

eek! Sorry that's just too much badly defined for my tast. I believe that a simple sketch would explain this much butter, something like (--p--). (I hope you understand this) Possibly have a group of sketches showing what is and what is not a neighbourhood.

'wiggle' or 'move' 2[edit]

I second that.

This "wiggle" or "move" definition is pretty shit.

It's necessary a great deal of competence in making abstract mathematical concepts intuitive. In some cases it can rather make it even harder to comprehend. ?The preceding unsigned comment was added by 200.164.220.194 (talk) 23:33, 6 March 2007 (UTC).

I changed "wiggle" to "perturb". It sounds more formal, but I am not sure it is better... Best, Sam nead 17:31, 29 May 2007 (UTC)

Which comes first: neighborhood of a point or of a set?[edit]

Oleg -- I looked through some books. The order is given below is the order with which I looked at them. "Counterexamples in topology" defines the neighborhood of a set and takes a point neighborhood as a special case. "General topology" by Kelly defines the neighborhood of a point and doesn't seem to mention the neighborhood of a set. "Topology, a first course" by Munkres defines neighborhoods of points and insists that neighborhoods be open. Spanier doesn't seem to mention neighborhoods at all. "Topology and geometry" by Bredon defines the neighborhood of a point. "Set theory and metric spaces" by Kaplansky defines the neighborhood of a point. Those are all of the reference that spring off the shelves.

So, if we have a bibliocracy, then we should remove the definition of the neighborhood of a set. Hmmm. I have to admit my preference for the way you suggested -- that we define the neighborhood of a set first and then take the neighborhood of a point as a special case. This way, we still present the "neighborhood of a point" idea, and retain the generality. I prefer this because it fits correctly with the notions of tubular neighborhood, regular neighborhood, and absolute neighborhood retract (ANR) in geometry and topology. I'll make the edits in a few minutes. best, Sam nead 16:55, 29 May 2007 (UTC)

My preference is actually for neighborhood of a point to be first, as that's the most fundamental definition, and your references seem to agree. The neighborhood of a set would then be an extension of this concept. Any strong feelings about this? :) Oleg Alexandrov (talk) 01:50, 30 May 2007 (UTC)
I think I have a definite preference for the way it is written, as of today. Let me try to justify this preference. 1) This way is slick. We get two definitions for the price of one. Now, "bogus generality" is something to be avoided, but this is not bogus generality because 2) the idea of a tubular or regular neighborhood of a submanifold is as important, if not more important, as the idea of the neighborhood of a point. At least, that is the case in geometry and topology: for example cut-and-paste arguments rely on tubular neighborhoods. Also, if you want to glue two manifolds-with-boundary in the smooth category, you have to build collar neighborhoods of the boundaries. (Hmm. Even then, the way you perform the gluing effects the smooth structure you get -- this is a step in Milnor's construction of exotic seven-spheres.)
Back on topic: 3) The references I gave above which didn't define the neighborhood of a set were textbooks. The reference which did define the neighborhood of a set was Counterexamples. That is not a textbook but rather is a reference book. As an encyclopedia, I think we are bound to resemble the latter not the former. Best, Sam nead 05:34, 30 May 2007 (UTC)
I find the "textbooks" vs "references books" argument unconvincing. Wikipedia articles should be as elementary as possible, so that people can learn things, see Wikipedia:Make technical articles accessible. So we should imitate textbooks, not reference books.
Also, defining the neighborhood of a point first does not interfere at all with tubular neighbourhoods and all that stuff, since we immediately define the neighbourhood of a set too.
Lastly, in analysis the neighbourhood of a point is used all the time, while neighbourhood of a set are used much more rarely. Oleg Alexandrov (talk) 15:14, 30 May 2007 (UTC)
I agree that articles should be accessible, but I do not agree that Wikipedia is more like a text book. An encyclopedia is a reference, almost by definition! Also, I won't be convinced by your last point without examples. Finally, may I suggest that you look through the book "Counterexamples in topology"? I think you will find it interesting. best, Sam nead 23:30, 31 May 2007 (UTC)
I strongly agree with Oleg here. Wikipedia is aimed at a general audience. Introductions should be intelligible to someone who does not already know the subject, to the extent possible. This means avoiding unnecessary generality. Also, for what its worth, the Encyclopedic Dictionary of Mathematics (MIT Press) defines neighborhoods in terms of a point.--agr 02:41, 1 June 2007 (UTC)
Agree with Oleg. Define it for points first, then sets (possibly even separately from the main definition). I think you'll run into problems, for instance, axiomatizing the space if all of your neighborhoods are set-based. (I'm not saying it's impossible, but I just don't think you can carry it off quite as smoothly.) Notions such as a neighborhood basis (etc.) are conventionally thought of in terms of points. Also, for what it's worth, even Bourbaki defines neighborhoods in terms of points. Cheers, Silly rabbit 02:54, 1 June 2007 (UTC)
I'll add another voice of agreement with Oleg. The neighbourhood of a point is far and away the more common usage in my experience, and it is easy enough to generalise this up to neighbourhoods of sets as needed. Given the nature of Wikipedia it seems sensible to start with the more common usage first. -- Leland McInnes 03:05, 1 June 2007 (UTC)
Gah! Where did all of you people come from? Ok, the point I find persusive here is the invocation of the MIT Encyclopedia and of Bourbaki. However, I don't buy this lowest common denominator business: an encyclopedia is a reference book, not a textbook. If a page of Wikipedia is too hard to read, then there should be obvious links to the necessary background. It seems impossible for every page to be free of prerequisites. Although I suppose that neighborhoods are pretty basic!
Next, that's two people who have said that point neighborhoods are important in analysis. I'd like some examples, please. My preference would be for examples not taken from first year calculus, by the way. Here -- I'll start: if f(z) is a function in the plane, analytic in a neighborhood of the point x, then f(z) agrees with its power series expansion (computed at x) inside of a small disk about x. Furthermore the power series converges absolutely inside the disk. All bets are off if f is not analytic in a neighborhood about x. (Ok, so I didn't manage to get out of the undergraduate syllabus. But at least that is a pretty result which mentions the concept.) Best, Sam nead 05:50, 1 June 2007 (UTC)
I agree with Oleg, an encyclopedia is indeed a reference, but it is generally accepted that the articles should build from the most basic and generally used case up to the more abstract and difficult concepts, I have heard the analogy of a pyramid, and I think that works pretty well. We are trying to inform the general reader first. This doesn't mean making the entire article LCD, just the beginning so it is accessible. --Cronholm144 06:39, 1 June 2007 (UTC)
Yes Wikipedia is a reference, but we are also admonished to make technical articles accessible, and in this case that means it will be more helpful to start with the more common and more easily intuitive case first. I'm not suggesting neighbourhoods of sets shouldn't be mentioned, merely that discussion of that can come afterwards. -- Leland McInnes 13:17, 1 June 2007 (UTC)
One more voice of agreement with Oleg. The neighborhood of a point is the first approach taken in every topology class I've ever been a part of. I will concede that there are textbooks which do not do this, but these are the exception, not the rule. Keep in mind, as well, that when a textbook chooses to discuss neighborhoods first—before the axioms for a topology are given—then an open set can be defined as a set that is a neighborhood for each of its points. While Wikipedia is not a textbook, I think it is proper to take the best pedagogy from textbooks to apply to our pages here for the benefit of our general readership. VectorPosse 07:22, 1 June 2007 (UTC)
Agreed completely. Wikipedia should not confuse unnecessarily. There is no such thing as neighborhood of a set in a Calculus course. Jmath666 20:27, 5 June 2007 (UTC)

Pictures[edit]

Oleg -- Those pictures are very cool! Thank you for adding them -- they really help the article! Question: how did you draw them? Request: could you add a similar picture to the tubular neighborhood page? This could be something as simple as a graph of a function in the plane, with a (green) shaded region around it. (I like green!) Best, Sam nead 05:39, 30 May 2007 (UTC)

I used Inkscape, it is free on both Windows and Linux. I could draw the picture you want, but perhaps you want to give it a try to take the picture I made as a base and modify it to do what you want, that way you may learn something new. :) Oleg Alexandrov (talk) 05:42, 30 May 2007 (UTC)
Fair enough. I'll take a look. Best, Sam nead 15:04, 30 May 2007 (UTC)

Comments[edit]

I think that the section "Significance of neighbourhoods in analysis of real functions", especially the sentence "Other notions of distance will (as they ought to) lead to the same results in analysis, if they are properly formulated." is flatly wrong. The example I am most familiar with is the quasi-conformal distortion of a continuous map. This, by definition, depends on the metrics on both the domain and range.

Perhaps the writer was thinking of the continuity of a map -- it is certainly true that metrics giving the same topology give the same notion of continuity. But this fails for more refined analytical concepts. (Other possible examples: BMO, exotic differentiable structures). If nobody objects I will trim the section, by quite a bit. Hmm. Advise me: am I expected to find out the identity of the writer and inform them of my intentions? Or should I just "Be bold"? Best, Sam nead 17:31, 29 May 2007 (UTC)

Ok, I've been bold. I thought about it, and I think I know what the writer of the "Significance" section was after. The point is that different metrics on a set may (or may not) define the same topology. When they do define the same topology then the metrics are called equivalent. Obviously, equivalent metrics give identical notions of continuity. However, as noted above, they need not have identical analytic properties.
I've deleted the entire discussion and I will check and make sure that the notion of equivalent metrics appears someplace more appropriate. Best, Sam nead 17:52, 29 May 2007 (UTC)

Regular neighborhood[edit]

The page Dehn twist refers to a 'regular neighborhood', which is also in the 'see also' sec'n of this article, but I cannot find a reference. Is it the same as 'Uniform neighborhood'? If not, can we add a section to this article explaining what 'regular neighborhood' is and delete the 'see also' (unless reg. neighborhood is a vast and interesting topic deserving of its own article). Thanks. Zero sharp 15:07, 19 July 2007 (UTC)

I think in this case it means symmetric around c.--Cronholm144 15:36, 19 July 2007 (UTC)
Regular neighborhood is indeed a big topic needing its own article. There are even "relative regular neighborhoods", which should be considered a "power steering" version of the vanilla regular neighborhood. To explain regular neighborhood, one would first need to specify a category (such as PL or DIFF) and then explain something like collapsing or whatnot. On the other hand, all that is meant in the Dehn twist article is taking a closed annulus containing the curve as its core. --C S 16:43, 13 September 2007 (UTC)