What is the difference between a ball and a neighbourhood?

Solution 1:

As you have seen, different texts define their terms somewhat differently, but the most common definitions are as follows:

• If $(X,d)$ is a metric space (or a pseudometric space), then the open ball of radius $r > 0$ about the point $x \in X$ is the set of all $y \in X$ such that $d(x, y) < r$.
• If $(X,d)$ is a metric space (or a pseudometric space), then a set $U \subseteq X$ is open iff for each $x \in U$ there is an $r > 0$ such that the open ball about $x$ of radius $r$ is a subset of $U$.
• An open neighborhood of a point $x$ in a metric space (or, in fact, any topological space) is any open set containing $x$.
• A neighborhood of a point $x$ in a metric space (or any topological space) is any subset of the space including, as a subset, an open neighborhood of $x$.

Beware: the notations used for open balls vary radically among texts, with almost all imaginable permutations of where the point goes, where the radius goes, and (in some cases) where the name of the metric goes.