What's the difference between a ball and a neighborhood?

[80past2]

They seem to be pretty similar. It seems like you could switch the two around and things would still make sense.

Although a ball is a set, and a neighborhood is a region, I guess? So is the set of all y in a neighborhood of x the ball?

 

[Fredrik]

The term "neigborhood" is defined differently by different authors. If I recall correctly, Rudin defines "open neighborhood of x"="open ball around x". (When I say "around x", I mean that x is the point at the center). I don't think that definition is very common however. Some authors (e.g. Munkres, if I recall correctly) define "neighborhood of x"="open set that contains x", and others (e.g. the person who wrote the Wikipedia article) define "neighborhood of x"="set that contains an open set that contains x". People who use the last definition also use the term "open neighborhood of x", defined as "neighborhood of x that's also an open set", or equivalently as "open set that contains x".

You are absolutely right that you can almost always replace "open ball around x" (or "open ball that contains x") with "open set that contains x", or vice versa. For example, the following definitions assign exactly the same meaning to the word "convergent":

1. A sequence is said to be convergent if there's a point x such that every open ball around x contains all but a finite number of terms.
2. A sequence is said to be convergent if there's a point x such that every open ball that contains x contains all but a finite number of terms.
3. A sequence is said to be convergent if there's a point x such that every open set that contains x contains all but a finite number of terms.

The reason for this is the simple relationship between open sets and open balls. Open balls are open sets. Open sets are unions of open balls.

 

[Deveno]

a neighborhood is a more general kind of concept than a ball.

a ball is one kind of neighborhood. there are other kinds, some of which give rise to different topologies, and some of which give rise to the same topology as open balls. for example, instead of balls, one can use open boxes, or open tetrahedra, all of which represent different ways of generalizing open intervals on the real line.

one thinks of neighborhoods as being sort of small fuzzy regions around a point. the idea of neighborhood makes more sense when you have a space where you have no way of measuring distances, so you use a neighborhood system to determine when a point is near a set.

one advantage to using neighborhoods instead of balls in sets (spaces) where you have no metric (distance function), is one can use neighboorhoods to define a filter, which can be thought of a a collection of sets that "zero in" on a point they are a neighborhood OF. filters allow one to define convergence and limits even when you don't have epsilons and deltas anymore.

since filters are very general, and neighborhoods themselves can be quite varied, it's usually easier to develop an intuition of how they behave by examining the specific kind of neighborhood, an epsilon-ball in Rⁿ, in some detail and using that as a guide.