4 ms·
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.wikip
by 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 agoI 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 agoBalls 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.