I'm self-studying Abbot's Understanding Analysis and I'm having trouble understanding $F_{\sigma}$ and $G_{\delta}$ sets. The question 3.5.2 is "Replace each ____ with the word finite or countable depending on which is more appropriate."
b) The ___ intersection of $F_{\sigma}$ sets is an $F_{\sigma}$ set.
c) The ___ union of $G_{\delta}$ sets is a $G_{\delta}$ set.
For b) I can set the problem up as:
$\bigcap\limits_{k=1}^{?}\bigcup\limits_{i=1}^{\infty} A_{i}^{k}$
I know that the finite union of closed sets is closed and that the arbitrary intersection of closed sets is closed, but I'm not sure if (or how) I would use that to determine that the finite intersection is a $F_{\sigma}$ set. This would also imply that the countable intersection of $F_{\sigma}$ sets is not always a $F_{\sigma}$ set.
For c) I can set the problem up as:
$\bigcup\limits_{k=1}^{?}\bigcap\limits_{i=1}^{\infty} A_{i}^{k}$
I think this problem follows similarly to b) hence I'm stuck here as well.
Best Answer
c) is same as b) with sets replaced by their complements. In c) the right word if 'finite'. Use the fact that $\bigcap U_i \cup \bigcap V_j=\bigcap_{i,j} (U_i\cup V_j)$.
It is well known that $\mathbb Q$ is not a $G_{\delta}$. But it is the union of singleton subbsets each of which is a $G_{\delta}$. So 'countable' fails in c).