A representation of $F_{\sigma\delta}$-ideals?

Question

First some definitions. By $\mathcal P(\mathbb N)$ we denote the family of all subsets of $\mathbb N$ endowed with the metrizable separable topology generated by the countable base consisting of the sets $[A;B]=\{C\subset \mathbb N:C\cap B=A\}$ where $A,B$ run over finite subsets of $\mathbb N$.

A subfamily $\mathcal F\subset\mathcal P(\mathbb N)$ is called hereditary if for any sets $F\in\mathcal F$ each subset of $F$ belongs to $\mathcal F$.

An ideal on $\mathbb N$ is a hereditary subfamily $\mathcal I\subset \mathcal P(\mathbb N)$, closed under unions.

An ideal $\mathcal I$ on $\mathbb N$ is called an $F_{\sigma\delta}$-ideal if $\mathcal I=\bigcap_{n=1}^\infty\bigcup_{m=1}^\infty\mathcal K_{n,m}$ for some closed subsets $\mathcal K_{n,m}$ of the compact topological space $\mathcal P(\mathbb N)$.

Problem 1. Can each $F_{\sigma\delta}$-ideal $\mathcal I\subset\mathcal P(\mathbb N)$ be written as $\mathcal I=\bigcap_{n=1}^\infty\bigcup_{m=1}^\infty\mathcal K_{n,m}$ for some hereditary closed subsets $\mathcal K_{n,m}$ of the compact topological space $\mathcal P(\mathbb N)$?

We can ask also a more general

Problem 2. Can each hereditary $F_{\sigma\delta}$-set $\mathcal I\subset\mathcal P(\mathbb N)$ be written as $\mathcal I=\bigcap_{n=1}^\infty\bigcup_{m=1}^\infty\mathcal K_{n,m}$ for some hereditary closed subsets $\mathcal K_{n,m}$ of the compact topological space $\mathcal P(\mathbb N)$?

Problem 2 is a related to another open (?)

Problem 3. Does each Borel hereditary subset $\mathcal F\subset\mathcal P(\mathbb N)$ belong to the smallest subfamily of $\mathcal P(\mathbb N)$ containing all open hereditary sets and closed under taking countable unions and intersections?

Answer 1

In [1], Solecki proved (Theorem 2.1 combined with Lemma 2.4),

Theorem: $\mathcal{I} \subset \mathcal{P}(\mathbb{N})$ is an analytic P-ideal, if and only if, there is some $\varphi:\mathcal{P}(\mathbb{N})\rightarrow [0,\infty)$ satisfying the following,

• $\varphi(\emptyset) = 0$,
• $\varphi(A) \le \varphi(A \cup B) \le \varphi(A) + \varphi(B)$, and
• $\varphi(A) = \lim_{n\rightarrow \infty} \varphi(A\cap n)$,

such that $\mathcal{I} = \{ A \subset \mathbb{N}: \limsup_{n} \varphi(A \backslash n)=0\}$. Furthermore, every analytic ideal on $\mathbb{N}$ is either a $P$-ideal, or finite-to-one reducible to the ideal generated by finite unions of the sets $\{ n\} \times \mathbb{N}.$

Remark: Any $\varphi$ which satisfies the hypothesis of the theorem is called a lower semi-continuous sub-measure on $\mathbb{N}$.

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An easy corollary to the previous theorem provides a partial answer to Problem 1.

Every $F_{\sigma\delta}$ $P$-ideal on $\mathbb{N}$ can be written as $\bigcap_n \bigcup_m K_{n,m}$ for some closed and hereditary $K_{n,m}$.

Proof: Using the previous theorem, it follows that for each $F_{\sigma\delta}$ P-ideal $\mathcal{I}$ there is a lower semi-continuous $\varphi$ such that $\mathcal{I} = \{ A \subset \mathbb{N} : \limsup_{n} \varphi(A\backslash n) = 0\}$. Letting $K_{n,m} = \{ A \subset \mathbb{N}: \varphi(A\backslash m) \le \frac{1}{n} \}$, we have $\mathcal{I} = \bigcap_n \bigcup_m K_{n,m}$; so it only remains to show that each $K_{n,m}$ is closed and hereditary.

To this end note that, for each $A \in K_{n,m}$ and $B\subset A$ we have $B\backslash m \subset A\backslash m$, therefore using the properties of $\varphi$, it follows that, $\varphi(B\backslash m) \le \varphi(B\backslash m \cup A\backslash m)=\varphi(A\backslash m) \le \frac{1}{n}$ and so $B \in K_{n,m}.$

Next, assume $A \subset \mathbb{N}$ has the property that for every $k \in \mathbb{N}$ letting $a_k = A \cap k$ and $b_k =k$, we have $[a_k, b_k]\cap K_{n,m} \neq \emptyset$. Then, for every $k \ge 0$, $\varphi(a_k \backslash m) \le \frac{1}{n}$ (since the sequence is increasing.) It immediately follows that $\varphi(A \backslash m) = \lim_{k} \varphi(a_k \backslash m) \le \frac{1}{n}$ and $A \in K_{n,m}$.

[1] Solecki, Sławomir, Analytic ideals and their applications, Ann. Pure Appl. Logic 99, No.1-3, 51-72 (1999). ZBL0932.03060.

Answer 2

Let $\mathcal{I}$ be an $F_{\sigma \delta}$-set (not necessarily hereditary) in $\mathcal{P}(\mathbf{N})$, i.e., there exist a family of closed sets $\{F_{n,m}: n,m \in \mathbf{N}\}$ such that $$ \mathcal{I}=\bigcup_{n\ge 1}\bigcap_{m\ge 1}F_{n,m}. $$ Since $\mathcal{P}(\mathbf{N})$ is a zero-dimensional space with countable base $\mathcal{B}=\{B_k\}$ of clopen hereditary sets (made of cylinders), then it is easy to see that each $F_{n,m}$ can be written as finite union of countable intersection of elements in $\mathcal{B}$. Hence there exist $\{B_{n,m}(i,j): i\ge 1, j=1,\ldots,k\}\subseteq \mathcal{B}$ such that $$ F_{n,m}=\bigcup_{j=1}^k \bigcap_{i\ge 1} B_{n,m}(i,j). $$ for all $n,m$. Therefore $$ \begin{align} \mathcal{I}&=\bigcup_{n\ge 1}\bigcap_{m\ge 1}\bigcup_{j=1}^k \bigcap_{i\ge 1} B_{n,m}(i,j)\\ &=\bigcup_{n\ge 1}\bigcap_{m\ge 1}\bigcap_{i_1\ge 1} \cdots \bigcap_{i_k\ge 1}\left(B_{n,m}(i_1,1)\cup \cdots B_{n,m}(i_k,k)\right). \end{align} $$ Since finite unions of hereditary sets are hereditary, the answer to Problem 2 is affirmative.