TITLE: Why use neighborhood to define boundary? Not open ball?
QUESTION [10 upvotes]: One way to define a boundary point of set S is that "every neighborhood of it contains at least one point of S and at least one point outside S". 
I wonder if it's OK to replace "neighborhood of it" by "open ball centered at it"? What's the difference? A further question is why introduce the concept "neighborhood" in the first place? Its role seems very similar to open balls.

REPLY [2 votes]: While the other answers cover the most obvious reason well, let me add a couple of reasons for defining and using the concept of neighborhoods that apply even in metric spaces:

Quite often, one needs to discuss sets that contain an open ball about a particular point, but are not open balls themselves. For example, "Let $S$ be a compact neighborhood of $x$" is easier to say repeatedly than "Let $S$ be a compact set that contains an open ball about $x$". If you need to refer to a concept enough, it is useful to have a special name for that concept. (Though I admit that many people restrict "neighborhoods" to be open, so this particular example does not apply to them - but there are other cases.)
Abstraction is one of the most important tools in the mathematician's toolbox. By being too specific in your definitions, you sometimes find yourself bogged down in needless details, while missing what is really important. In your example of the definition of the boundary point, the only thing that is important for that definition is no matter how close you look around the point, you see points that are inside and that are outside. The particular shape of how you are looking around the point makes no difference. So a definition that does not require a particular shape helps you to see what is important.

The latter reason may not make much sense to you without more experience, but I cannot tell you how many times I've struggled mightily to prove something, only to come back to it a few years later and discover that it is almost trivial from the right point-of-view, and all the details I fought through were unnecessary.