TITLE: Why are finite intersections (cylinder sets) of this form?
QUESTION [1 upvotes]: I am referring to this article about cylinder sets.
My questions concern the section "Definition for infinite products of finite, discrete sets".
1.) The $k$-th projection maps the $k$-th entry $x_k$ of the bi-infinite sequence $x$ onto the map containing $x_k$, i.e.
$$
p_k\colon \left\{0,1,...,n-1\right\}^{\mathbb{Z}}\to \left\{0,1,...,n-1\right\}, x\mapsto \left\{x_k\right\}?
$$
Since given any open set in $\left\{0,1,...,n-1\right\}$, f.e. the subset $\left\{0,1\right\}$, what would be the preimage under $p_k$? This only seems to make sense for subsets containing only one letter. This is why I ask how the $k$-th projection looks like.
2.) Why are intersections of finite many open cylinders (what the text then calls cylinder set) of the form
$$
C_t[a_0,...,a_m],
$$
where the positions are from $t$ to $t+m$, i.e without any gaps between them?
What is for example with the intersection $C_{-10}[0]\cap C_0[2]\cap C_{3}[2]$? This is an intersection of finite many open cylinders but the prescribed positions have gaps between each other...

REPLY [1 votes]: The preimage of $\{0,1\}$ under $p_k$ is 
$$\left\{\langle x_m:m\in\Bbb Z\rangle\in\{0,\ldots,n-1\}^\Bbb Z:x_k\in\{0,1\}\right\}\;,$$
the set of all doubly infinite sequences whose terms come from the set $\{0,\ldots,n-1\}$ and whose term indexed by $k$ is $0$ or $1$. In the cylinder set notation that you’re using it is $C_k[0]\cup C_k[1]$.
$C_{-10}[0]\cap C_0[2]\cap C_3[2]$ is the set of sequences $\langle x_m:m\in\Bbb Z\rangle\in\{0,\ldots,n-1\}^{\Bbb Z}$ such that $x_{-10}=0$ and $x_0=x_3=2$:
$$C_{-10}[0]\cap C_0[2]\cap C_3[2]=\left\{\langle x_m:m\in\Bbb Z\rangle\in\{0,\ldots,n-1\}^\Bbb Z:x_{-10}=0\text{ and }x_0=x_3=2\right\}\;.$$
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