Understanding the Difference Between Open Balls and Neighborhoods

[mynameisfunk]

Are open balls and neighborhoods the exact same thing? If not, could you please shed some light on this for me?

 

[Office_Shredder]

A neighborhood is just an open set. An open ball requires being in a metric space. The word neighborhood is usually used as opposed to just open set because you want to give the impression that the open set is supposed to be a small one, similar to saying let $$\epsilon>0$$ vs saying let $$M>0$$. They both say the exact same thing but one of them indicates we're interested in picking small numbers and one large numbers. It's not a formal definition but just to give the reader some intuition

 

[Fredrik]

There are at least three inequivalent definitions of "neigborhood of x":

1. An open ball around x.
2. An open set that contains x.
3. A set that has an open subset that contains x.