Talk:Ball (mathematics)

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 Field:  Geometry

Overline for closed ball[edit]

I object to the use of the bar to denote closed ball. While I know this is a very old and traditional notational convention, it's one that should be stopped, for a simple reason. The bar is also commonly used to denote topological closure. Even in very simple metric spaces, these two notations conflict. Take X = {0, 1} with disrete metric. Then B(0, 1) = {0}, i.e. open ball of radius 1 about the point 0 is just the singleton {0}. The "closed ball of radius 1 about the point 0" B-bar(0, 1) = {0, 1} = X, yet the "closure of the open ball of radius 1 about the point 0" is just {0}, not X. Some notation that indicates < or =<, or which uses B and D instead of just B and B-bar, would be more clear. This is not mere nit-picking — this issue comes up all the time in p-adic analysis. Revolver 09:39, 29 Oct 2004 (UTC)

That's a good point, but the notation is at least suggestive. I think an analogy can be made to the term "manifold with boundary" — here boundary rarely denotes the topological boundary. I never really liked the use of B vs. D either: disk tends to suggest a 2-dimenisonal object, while ball suggests a 3-dimensional one. As to why one should naturally have a boundary and the other not is beyond me. -- Fropuff 15:28, 2004 Oct 29 (UTC)
That's just my point — the notation is suggestive, and is suggestive of something which is false in many cases — namely, that the closure of an open ball is always the closed ball. (The only difference is the bar is only over the B instead of the whole thing.) This is often not the case in ultrametric spaces. "Manifold with boundary" isn't notation, it's a definition, (much like "multi-valued function") so I don't see the comparison. I never liked B or D either, in the sense you mention, for the same reasons, but here the suggestion misleads against intuition but doesn't actually conflict with other notation in the way that B-bar does. Revolver 23:12, 29 Oct 2004 (UTC)
I agree to the objection. The overline should be used to denote the closure. I suggest an index (subscript) "c" for the closed ball (and maybe optionally 'o' for the open ball). The problem arises if people want to write the radius as subscript. As long as the notation B(x;r) is used, the only possible confusion is between the point and the radius, as some would rather write B(r;x)... MFH: Talk 14:32, 22 Apr 2005 (UTC)

You could write it as a superscript: Bc for closed balls, and B or Bo for open ones. The question is whether this notation is standard; we shouldn't really be inventing notation here. I've seen both the bar notation and the B vs. D notation, neither of which is very good for the reasons mentioned above. -- Fropuff 16:09, 2005 Apr 22 (UTC)

I never saw the superscript, while the subscript is quite standard AFAIK. Of course the overline is very frequent, this is the standard notation for closure, and in the usual case, this coincides with "≤".
Putting the radius as index is handy in quick handwriting when doing calculations (I also do it), but when typesetting it, the B(centre, radius) notation is preferable for many reasons, IMHO. MFH: Talk 13:14, 25 Apr 2005 (UTC)

I argue that the overline notation is correct, because the closed ball is indeed the closure of the open ball. When this fails to be equivalent to "all points of distance ≤ r", it is still the case that the closed ball is the closure of the open ball. That is, in these situations, the closed ball cannot be deemed equivalent to "all points of distance ≤ r". In particular, the reason the overline notation is popular and the superscript "c" is unknown is that the former is actually correct. 𝕃eegrc (talk) 18:15, 9 December 2015 (UTC)

Ball as nbhd[edit]

An (open) ball is any open set: one speaks of "a ball about the point p" when one means an open set containing p. What this ball is homemorphic to depends on the ambient space and on the ball chosen. A closed ball is the closure of a ball. Neighborhood (or neighbourhood) is sometimes used instead of ball, although neighborhood also has a more general meaning: a neighborhood of p is any set containing a ball about p.

I've never heard the term "ball" used in this way in topology, always in reference to metric spaces. I've heard the term "neighbourhood" used both for what is called "ball" and "neighbourhood" here, with usage varying. I don't remember hearing "ball" used for top spaces much, though. Revolver 09:44, 29 Oct 2004 (UTC)

I've not heard this usage before either. At the best, it is a very informal language. This should be emphasized if this section is to remain. The proper language is (open/closed) neighborhood. -- Fropuff 15:28, 2004 Oct 29 (UTC)
I think it might be an informal term used in Euclidean space or manifolds, where in essence you can say open set = ball, since balls form local base. Revolver 23:12, 29 Oct 2004 (UTC)
Fine: I've edited the Topology section somewhat, per Fropuff's suggestion. See if y'all like it better now.msh210 18:41, 9 Nov 2004 (UTC)
I have never seen this usage before either, not even in any informal way, and it is really misleading: if every open set (containing a point) is a ball, then for instance an annulus is also a ball! I think it should be deleted. 08:39, 16 October 2005 (UTC)

Geometry and topology[edit]

The recent changes by Patrick took the "Geometry" header away from the discussion of geometry; the "Topology" header remains above the topology discussion. To me this is blatant antitopologinarianism.  :-)  It implies that the ball of geometry is the real ball, whereas the ball of topology is "merely" a topological ball (and so gets put under a special header). I think we need to bring back the "Geometry" header.msh210 14:29, 16 August 2005 (UTC)

I fixed the headers. I put geometry under metric space because the same formula can be used, it does not become simpler for Euclidean space.--Patrick 15:01, 16 August 2005 (UTC)

Inside of a sphere or octahedron ?[edit]

Those two defs seem at odds with each other, to me (I suppose an octahedron can be placed inside a sphere, but so could many other shapes). I've added a link from Ball (disambiguation), describing it as a multidimensional sphere. If you have a better def for the disambiguation page, please change it. StuRat 07:07, 11 December 2006 (UTC)

Which two defn's are you referring to? An octahedron is a ball under the L1-norm on R3, whereas an "ordinary ball" is a ball under the Euclidean (or L2) norm. -- Fropuff 17:00, 11 December 2006 (UTC)

Fake 3-ball[edit]

Fake 3-ball, what's that? --Abdull (talk) 23:03, 13 January 2010 (UTC)

I am not sure if the terminology is completely standard, but as I remember a "fake 3-ball" is a compact contractible 3-manifold M which is not homeomorphic to the standard closed 3-ball (maybe some definitions also include the condition that the boundary of M is homeomorphic to the 2-sphere, I don't remember for sure). The Poincare conjecture, which is now a theorem, implies that a fake 3-ball does not exist and that every compact contractible 3-manifold is in fact homeomorphic to the standard closed 3-ball. That is probably why the name has been redirected here. Nsk92 (talk) 15:05, 14 January 2010 (UTC)

Use of "interior"[edit]

There are several instances in the current article where the ball is referred to as the "interior" of the sphere. Unfortunately "interior" also means something distinct in a context very near this one; specifically the (topological) interior of any set is the largest open set contained in that set. In particular, the (topological) interior of the closed ball is the open ball. Importantly, the (n−1)-dimensional sphere interpreted as embedded n-dimensional space has the empty set as its (topological) interior. Instead interpreting the (n−1)-dimensional sphere as its own manifold gives the entire (n−1)-dimensional sphere as its own (topological) interior. Either way, the (topological) interior of the sphere is not the ball. Is anyone else concerned about this? Should we change the article to either avoid the use of "interior" or to clarify that it is not the topological interior that is meant? 𝕃eegrc (talk) 14:08, 9 December 2015 (UTC)

Assessment comment[edit]

The comment(s) below were originally left at Talk:Ball (mathematics)/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

This is an important topic, but the current treatment is back-to-front and unencyclopedic. The article should start with the ball in Euclidean 3-space, then move on to more general situations. Subtle issues such as whether the closure of an open ball is a closed ball should not dominate the article. Geometry guy 13:41, 13 May 2007 (UTC)

Last edited at 13:41, 13 May 2007 (UTC). Substituted at 01:47, 5 May 2016 (UTC)