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?