A set $A ⊆ R$ is called an $F_\sigma$ set if it can be written as the countable union of closed sets. A set $B ⊆ R$ is called a $G_δ$ set if it can be written as the countable intersection of open sets.
How to prove, that:
- The countable union of $F_σ$ sets is an $F_σ$ set.
- The finite intersection of $F_σ$ sets is an $F_σ$ set.
Thanks for help!
Suppose $P_n, n \in \mathbb{N}$ are $F_\sigma$ sets so we can write $$P_n = \bigcup_{m \in \mathbb{N}} F_{n,m},\text{ with } F_{n,m} \text{ closed in} X$$
But then $$\bigcup_{n \in \mathbb{N}} P_n = \bigcup_{(n,m) \in \mathbb{N} \times \mathbb{N}} F_{n,m}$$
and the union is also a countable (as $\mathbb{N} \times \mathbb{N}$ is countable) union of closed sets, hence an $F_\sigma$.
Also $$P_1 \cap P_2 = \bigcup_{(n,m) \in \mathbb{N} \times \mathbb{N}} (F_{1,n} \cap F_{2,m})$$
(check this identity), so the intersection of two $F_\sigma$'s is an $F_\sigma$ again, and by induction we can extend this to all finite intersections in the standard way, or alternatively use that
$$\bigcap_{i=1}^k P_i = \bigcup_{f \in \mathbb{N}^k} \bigcap_{n=1}^k F_{n,f(n)}$$ as a generalisation of the previous identity.