Places of a number field I am reading the book Class Field Theory by Nancy Childress. In the chapter on Idelic theory, there is the following definition.

Let $F$ be an algebraic number field. An absolute value on $F$ is a mapping 
  $$\|\text{ }\|:F\rightarrow [0,\infty)$$
  that satisfies $\|0\|=0$ and the restriction of $\|\text{ }\|$ to $F^{\times}$ is a homomorphism of multiplicative groups $F^{\times}\rightarrow\mathbb{R}_{+}^{\times}$ and that satisfies 
  $$\|1+x\|\leq c\text{ whenever } \|x\|\leq 1$$
  for some suitable constant $c$.

Then it is said that this absolute value induces a topology on $F$ via a fundamental system of neighborhoods of the form 
$$\lbrace x\in F:\|x-a\|<\epsilon\rbrace, \text{ }\epsilon>0$$ 
I don't know what $a$ is in the above definition. Is it any arbitrary point of $F$ ?
 A: You are right that $a$ can be any element of $F$, but the idea is that we are forming an open cover of $F$ (topologically speaking), where the amount we require is dependent on $\epsilon$.
As an example, take $F=\mathbb{Q}$ with the usual (archimedan) absolute value. If $\epsilon=1$, then we can taking the set $\{k/2 \, : \, k \in \mathbb{Z} \}$ forms an open cover (note strict inequality) so the system of $a$'s that we choose are of the form $k/2$ and indeed any rational is within $1$ of these points.
For a slightly more difficult example, consider the $p$-adic absolute value instead and let $\epsilon=1$ again. Take the set $\{ a=\sum_{i=-N}^{0} a_ip^i \, : a_i \in \{0,..,p-1\}, N \geq 0 \}$ as your system of $a$'s this time. We shall show this does form an open cover.
Let $x=\sum_{i=-N}^{M} x_ip^i$ and take $a=\sum_{i=-N}^{0} x_ip^i$. Then $|x-a| = |\sum_{i=1}^{M} x_ip^i| \leq p^{-1} <1$ by the ultrametic triangle inequality. (In fact the valuation is $p^k$ where $k>0$ is the smallest positive integer such that $x_k \neq 0$.)
Lastly, I wish to remark that whilst I have constructed such neighbourhoods, these are by no means unique; I could add more possible values of $a$ or shift them all by some random constant.
