Let \(F, F_{1}, F_{2}, F_{3}, \ldots\) be \(F_{\sigma}\)-sets in \(\mathbb{R}^{n}\). Show that

(i) \(F_{1} \cap F_{2} \cap \cdots \cap F_{N}\) is for every \(N \in \mathbb{N}\) an \(F_{\sigma}\)-set;

(ii) \(\bigcup_{n \in \mathbb{N}} F_{n}\) is an \(F_{\sigma}\)-set;

(iii) \(F^{c}\) and \(\bigcap_{n \in \mathbb{N}} F_{n}^{c}\) are \(G_{\delta}\)-sets;

(iv) all closed sets are \(F_{\sigma}\)-sets.