What is the difference between open ball and neighborhood in real analysis?

Question

I'm learning real analysis.

Open ball: The collection of points $x \in X$ satisfying $|x - x_{0}| < r$ is called the open ball of radius $r$ centered at $x_{0}$

Neighborhood: A neighborhood of $x_{0} \in X$ is an open ball of radius r > 0 in $X$ that is centered at $x_{0}$

I'm using Real and Complex Analysis written by Christopher Apelian and Steve Surace. In my mind, open ball = a collection of points satisfy certain requirement = neighborhood. I do not find out any differences between open ball and neighborhood. Could any one explain it? Thanks!

Answer 1

Let's discuss the 1-dimensional case. An open ball in $\Bbb R$ is a set given by $$ B(x,r):=\{y\in \Bbb R:|y-x|<r\}. $$ These sets are very important as they allow us to define the topology on $\Bbb R$, i.e. it allows us to say which sets are open and which are not. Topology is one of the key structures to work with uncountable spaces, $\Bbb R$ in particular.

The neighborhood of a point $x\in \Bbb R$ is any subset $N_x\subseteq \Bbb R$ which contains some ball $B(x,r)$ around the point $x$. Note that in general one does not ask neighborhood to be open sets, but it depends on the author of a textbook you have in hands.

For example, if $x = 1$ then $(0.5,1.5)$ is a ball (of a radius $0.5$) around $x$, and $(0.5,1.5)\cup [2,4]$ is a neighborhood of $x$ which is not a ball (neither around $x$, nor around any other point).

As Brian has mentioned, indeed in your case these definitions are equivalent - but this is an unusual way to define neighborhoods.

Answer 2

An open ball about x is a ball about a point $x$. A neighborhood of $x$ is commonly defined as an set containing $x$ in its interior.

A neighborhood need not be a ball/other trivial shape just a set containing $x$.