# Open Balls and Neighborhoods

1. Sep 11, 2010

### mynameisfunk

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

2. Sep 11, 2010

### Office_Shredder

Staff Emeritus
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

3. Sep 11, 2010

### Fredrik

Staff Emeritus
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.