I read from "S. Solecki, Analytic ideals and their applications, Ann. Pure Appl. Logic, 99 (1999),51–71."
In the first; I can't understand what is the analytic ideal? I know what the ideal is but I don't know what the analytic ideal is.My second question:In page 53:
Let $\mathcal I$ be an ideal on $\omega $. A set $A \subset 2^\omega$ is called small if there exists $X \in \mathcal I $ such that$\{ Y \cap X: Y \in A\}$ is meager in $2^X$. Recall that a subset $A$ of $2^X$ is hereditary if subsets of elements of $A$ are themselves in $A$. Define$C(\mathcal I) = \{K \subset 2^X: K \text{ hereditary, compact, and such that } \forall X\in \mathcal I, \exists n \in \omega : X/n \in K\}$.
Where $X/n=X-\{0,1,...,n-1 \}$
Is the metric on $2^X$ is the same metric that on cantor space?
The cantor space is the set $2^\omega$ with the following metric:
$$ d(a_n,b_n)= \left\{ \begin{array}{lr} \frac{1}{m+1} & \mbox{if } m=\min\{k : a_k \neq b_k\} \\ 0 & \mbox{if } \{a_n\} = \{b_n\} \end{array}\right.$$
My third question: From the same paper:
Lemma $2.2.$ Let $K \subset 2^\omega $ be hereditary and compact, Then $K \in C(\mathcal I)$ if, and only if; $K$ is not small.
Proof. $\Rightarrow :$ is clear. To see $\Leftarrow :$ let $K$ be a compact hereditary set which is not small.Let $X \in I$. Then $\{X \cap Y: Y \in K\}$ is not meager in $2^X$. Since this set is compact, its interior in $2^X$ is non-empty. Since it is also hereditary, it is not difficult to see that there is $m$ such that $2^{X/m}$ is contained in it. Thus $X/m \in K$.
I cant understand the proof of $\Rightarrow $ it isn't clear at all for me.Also in the proof he said that $\{X \cap Y: Y \in K\}$ is compact. But we just know that $K$ is compact. How did he say that this set is compact?