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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.

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The sets are indexed by $\bigcup_{k=1}^\infty \mathbb N^{k}$. The idea is that each of the sets of "level" $k$ is divided into countably many subsets to give the sets of level $k+1$. Probably the best way to think of the index set of a sieve is to think of a rooted tree where each node $(n_1,\ldots,n_k)$ has countably many successors $(n_1,\ldots,n_k,l)$, $l \in \mathbb{N}$. –  t.b. May 20 '12 at 10:00
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Two good books that treat these ideas more in-depth are Kechris's Classical Descriptive Set Theory and Moschovakis's book available here. –  t.b. May 20 '12 at 10:03
    
Do you know what a sheaf is? –  Simon Markett May 20 '12 at 10:06
    
@t.b.: Thanks; this makes perfect sense then. How about the first condition. This means that the first 'level' sets cover $X$? –  Thomas E. May 20 '12 at 10:09
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Yes, the first-level sets cover $X$. –  Brian M. Scott May 20 '12 at 10:10

1 Answer 1

up vote 5 down vote accepted

I think that with Arveson's chapter 3 you've already found the maximally nice and efficient exposition of the basic theory of standard Borel spaces. Much more in-depth are Kechris's Classical Descriptive Set Theory and Moschovakis's freely available book on Descriptive Set Theory.

The fundamental idea is this: everything is going to be encoded by the “Baire space” $\mathcal{N} = \mathbb{N}^\mathbb{N}$ of integer sequences, equipped with the product topology. This is the “mother of all Polish spaces”, and it has a natural sieve of clopen sets, namely $A_{n_1 \cdots n_k} \subset \mathcal{N}$ is the set of integer sequences with initial segment $n_1,\ldots,n_k$. A sieve in a general space is then an axiomatization of two of the fundamental properties of the sets $A_{n_1\cdots n_k}$ in $\mathcal{N}$, namely the two properties you mention in your question.

As I said in a comment (with a typo), the index set of a sieve is $\bigcup_{k=0}^\infty \mathbb{N}^k$ which is best thought of as a tree with the empty sequence as a root node and every node $(n_1,\ldots,n_k)$ has countably many children $(n_1,\ldots,n_k,l)$, $l \in \mathbb{N}$. The properties of a sieve say that the sets of level $1$, the family $\{A_{l}\}_{l\in\mathbb{N}}$, covers $X$. Then each of the sets $A_{n_1}$ is divided into countably many subsets $\{A_{n_1 l}\}_{l \in\mathbb{N}}$ whose union is $A_{n_1} = \bigcup_{l=1}^\infty A_{n_1 l}$ and so on: each set $A_{n_1\cdots n_k}$ is divided into countably many subsets $A_{n_1\cdots n_k l},$ $l \in \mathbb{N}$.

If you start with a separable, complete metric space $X$ and your sieve is comprised of sets $A_{n_1\cdots n_k}$ which are closed balls with radii $\leq 1/k$ then every infinite sequence of integers $(n_1,n_2,\ldots)$ will specify the unique point $x \in \bigcap_{k=1}^\infty A_{n_1\cdots n_k}$ and this gives you a continuous surjection $\mathcal{N} \to X$. As you will see when reading further, such surjections will be at the very heart of everything that follows.

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It's amazing how clear this is now. Thanks. And about reading the book further: I can't wait :-) –  Thomas E. May 20 '12 at 11:09
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Great, that's very nice to hear :) Kechris is one of my favorite books in general and as a consequence on the topic in particular because it contains so many beautiful applications and exercises. One book that I forgot to mention in my answer is Srivastava's Course on Borel sets which is also nice. –  t.b. May 20 '12 at 11:42
    
Sure, I'll take a look on both of them. Thanks once again, this was really helpful. –  Thomas E. May 20 '12 at 11:49
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Besides Kechris and Srivastava, you may also find the following useful (listed in random order): Donald Cohn's Measure Theory (see Chapter 8: Polish Spaces and Analytic Sets); Kuratowski's Topology, Volume I (see the second half of the book); Sierpinski's General Topology (see the second half of the book); Bruckner/Bruckner/Thomson's Real Analysis (see Chapter 11: Analytic Sets). –  Dave L. Renfro May 20 '12 at 12:32
    
In addition, useful phrases to google are the phrases "Luzin sieve" and "Lusin sieve" (google separately) and the two word collections {"Luzin" AND "cribles"} and {"Lusin" AND "cribles"}. –  Dave L. Renfro May 20 '12 at 12:33

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