I'm reading William Arveson's book 'an invitation to C*-algebras' and I came across the concept of sieves: they seem to play an important role in Borel structures. However, I'm having hard time understanding what William is trying to say here in the basic definition, and I can't find this concept from any other book in my shelf. I tried Munkres, Engelking and Willard.
So here's basicly the definition word-by-word:
Let $X$ be a topological space. For every $k\geq 1$ and every $k$-tuple of positive integers $n_{1},...,n_{k}$, let $A_{n_{1}n_{2}\cdot\cdot\cdot n_{k}}$ be a subset of $X$. The family $\{A_{n_{1}n_{2}\cdot\cdot\cdot n_{k}}\}$ is called a sieve for $X$, if the following properties are satisfied: \begin{align*} &(i)\,\,\bigcup_{n_{1}=1}^{\infty}A_{n_{1}}=X\\ &(ii)\,\,\bigcup_{l=1}^{\infty}A_{n_{1}n_{2}\cdot\cdot\cdot n_{k}\,l}=A_{n_{1}n_{2}\cdot\cdot\cdot n_{k}}, \,\,\mathrm{for}\,\,\mathrm{every}\,\,k\geq 1\,\,\mathrm{and}\,\,\mathrm{for}\,\,\mathrm{every}\,\,n_{1},...,n_{k}\geq 1 . \end{align*}
(And the sieve is called an open sieve if its a collection of open sets.)
I'm having hard time grasping this definition and interpreting what he is going after with it: what subsets does he actually choose into the sieve? Is he using the same index set $\{n_{1},n_{2},...\}$ which is basicly $\mathbb{N}$ with a different order? Are the sets being indexed by a "product-index" of each $k$-tuple of indices?
Thanks for all the input in advance.