I am trying to understand a family of types $\mathcal{F}$ in the set $S(A)$,the set of complete types over $A$ which is a compact and Haurdorff space equipped with the logic topology. The family $\mathcal{F}$ turns out to be a $G_{\delta}$ set. Hence the following questions came up:

1. What is know about the subspace topology induced over a $G_{\delta}$ set?

2. Is there any possible topology that one could induce over $S(A)$ that makes and $F_{\sigma}$ set closed?

3. Which properties of convergence could be stated among sequences in a $G_{\delta}$ set?

Any comment, help or reference will be highly appreciated.