Precise definition of epsilon-ball My textbook gives the following definition: "For each $\epsilon>0$, the $\epsilon$-ball about a point $x$ in a metric space $M$ is the set $\{y\in M:d(x,y)<\epsilon\}$."
Is this correct? Because this sounds as if $\epsilon$ is just a dummy variable, and that there is such a thing as, say, a "5-ball" meaning $\{y\in M:d(x,y)<5\}$.
Shouldn't the correct definition be something along the lines of "The $\epsilon$-ball about a point $x$ in a metric space $M$ is the set $\{y\in M:$ for each $\epsilon >0, d(x,y)<\epsilon\}$." ?
 A: No; in fact, the set you mention consists only of the point $x$, which is certainly not what we want! 
Perhaps better is to define an $\epsilon$-ball, rather than the $\epsilon$-ball. Each $\epsilon$ ball is different if $\epsilon$ is different. 
You're right that the $\epsilon$ is somewhat arbitrary. Usually what we want to say is something like, "The set $S$ contains an $\epsilon$-ball around $x$," where we don't care what the precise $\epsilon$ is.  The key is that it can be as small as you like. 
Here's a good example: take a point $p$ in the plane which is on the $y$-axis, and let $S$ be the right half plane, i.e. $\{ (x,y)\in \mathbb{R}^2 \mid x>0 \}$. Then we can say that for all $\epsilon>0$, there is a point $s\in S$ such that $d(p,s)<\epsilon$. Any particular point in $S$ will be a finite distance away, but the point is that $S$ is as close to $p$ as we like. Another way of saying this is, "$x$ is in $S$-closure."
$\epsilon$ is used as a sort of shorthand. When someone says "$\epsilon$" in mathematics, everyone knows, "oh, she means a positive number that we can think of as arbitrarily small". Of course there is no single number that is arbitrarily small, but usually we are talking about a condition that holds for all $\epsilon >0$. So no mattter how small you go, the condition still holds. This is the case when we talk about limits and continuity, for example.
When you study metric spaces (if you haven't already), you'll learn that you can give $\mathbb{R}^n$ many different norms. In these norms, the distance between two points may be something different than in the norm we're used to. And yet, all the definitions of continuity, limits, etc. stay the same! How can that be? It is because these norms are equivalent, meaning that, although all the individual distances are different, they both get arbitrarily small in the same places. So for instance, if you have a sequence going to zero in one norm, it will still go to zero in the other norm. So all the definitions using "$\forall \epsilon>0$" still hold in any norm you like!
A: Your first impression is correct. There is a $5$-ball and a ${\sqrt 2}$-ball and a $0.000001$-ball. Your second suggestion would amount to the singleton set $\{x\}$.
