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> 1. A neighborhood of a point p in a Euclidean space is the set of all q such that d(p, q) < r, where r is some radius greater than 0. Isn't that what mathema

by alimw 9y ago

> 1. A neighborhood of a point p in a Euclidean space is the set of all q such that d(p, q) < r, where r is some radius greater than 0. Isn't that what mathematicians call a ball rather than a neighbourhood? A name which rather suggests that they are visualising it.

dsacco 9y ago
Not really, no. It’s a neighborhood. A ball is one description of a neighborhood in a specific context. I hesitate to link to Wikipedia, but: https://en.m.wikipedia.org/wiki/Neighbourhood_(mathematics) https://en.m.wikipedia.org/wiki/Neighbourhood_(mathematics) It would not be correct to use terminology like “ball” to describe a “neighborhood”, even if, in specific instances, they are the same representation in different contexts (i.e. geometric vs analytic).
- alimw 9y ago
  I am not using 'ball' to describe a neighbourhood, you are using 'neighbourhood' to describe a ball! From your own link: In a metric space M=(X,d), a set V is a neighbourhood of a point p if there exists an open ball with centre p and radius r>0, such that B_r (p)=B(p;r)={x∈X | d(x,p)<r} is contained in V. (My emphasis.)
- xelxebar 9y ago
  Balls are definable in metric spaces, neighborhoods in general topological ones, so "neighborhood" is indeed the more general term. That said, once we have a metric space, the metric balls give a natural basis for the topology, which means that every theorem about neighborhoods has a corresponding one that uses balls instead. So in practice, when in a metric space we just use balls everywhere because they let us exploit their extra structure if it's needed. This is why you see the abuse of language calling neighborhoods balls, despite that being technically incorrect.
philipov 9y ago
A sphere is the surface of a ball. Your criticism applies to (2) but not to (1). However, 'ball' is from geometry, while 'neighborhood' is from analysis. They overlap, but they're not to be confused, because 'neighborhood' is used to emphasize that one should not visualize it.
- dsacco 9y ago
  > However, 'ball' is from geometry, while 'neighborhood' is from analysis. They overlap, but they're not to be confused, because 'neighborhood' is used to emphasize that one should not visualize it. You’re right, but in fairness I’ve seen multiple analysis textbook authors use terms like “circle” and “ball” to describe a neighborhood in topology. This is usually in the fashion, “...(in other words, E is a ball.” This is precisely the sort of thing I dislike, because aside from being technically incorrect like you say, it can (for that section) encourage a student to think, “Oh, why didn’t he just say that?” and skip the heavier definition -> theorem -> proof sequence preceding it, to the detriment of understanding later material.

by alimw 9y ago

dsacco 9y ago

Not really, no. It’s a neighborhood. A ball is one description of a neighborhood in a specific context. I hesitate to link to Wikipedia, but: https://en.m.wikipedia.org/wiki/Neighbourhood_(mathematics) https://en.m.wikipedia.org/wiki/Neighbourhood_(mathematics) It would not be correct to use terminology like “ball” to describe a “neighborhood”, even if, in specific instances, they are the same representation in different contexts (i.e. geometric vs analytic).

alimw 9y ago
I am not using 'ball' to describe a neighbourhood, you are using 'neighbourhood' to describe a ball! From your own link: In a metric space M=(X,d), a set V is a neighbourhood of a point p if there exists an open ball with centre p and radius r>0, such that B_r (p)=B(p;r)={x∈X | d(x,p)<r} is contained in V. (My emphasis.)
xelxebar 9y ago
Balls are definable in metric spaces, neighborhoods in general topological ones, so "neighborhood" is indeed the more general term. That said, once we have a metric space, the metric balls give a natural basis for the topology, which means that every theorem about neighborhoods has a corresponding one that uses balls instead. So in practice, when in a metric space we just use balls everywhere because they let us exploit their extra structure if it's needed. This is why you see the abuse of language calling neighborhoods balls, despite that being technically incorrect.

philipov 9y ago

A sphere is the surface of a ball. Your criticism applies to (2) but not to (1). However, 'ball' is from geometry, while 'neighborhood' is from analysis. They overlap, but they're not to be confused, because 'neighborhood' is used to emphasize that one should not visualize it.

dsacco 9y ago
> However, 'ball' is from geometry, while 'neighborhood' is from analysis. They overlap, but they're not to be confused, because 'neighborhood' is used to emphasize that one should not visualize it. You’re right, but in fairness I’ve seen multiple analysis textbook authors use terms like “circle” and “ball” to describe a neighborhood in topology. This is usually in the fashion, “...(in other words, E is a ball.” This is precisely the sort of thing I dislike, because aside from being technically incorrect like you say, it can (for that section) encourage a student to think, “Oh, why didn’t he just say that?” and skip the heavier definition -> theorem -> proof sequence preceding it, to the detriment of understanding later material.