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I am trying to understand a family of types $\mathcal{F}$ in the set $S(A)$,the set of complete types over $A$ (in the sense of types in model theory) 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.

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    $\begingroup$ The notation $S(A)$ could mean anything. If you don't explain your notation no one will be able to answer. $\endgroup$
    – Nik Weaver
    Commented Aug 24, 2019 at 22:46
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    $\begingroup$ Good, thank you. It could also help in parts 1 and 3 if you gave some idea of what you're looking for, or an example of the kind of answer you would want ... otherwise these questions are too open-ended. $\endgroup$
    – Nik Weaver
    Commented Aug 25, 2019 at 3:07
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    $\begingroup$ @AndrejBauer en.wikipedia.org/wiki/Type_(model_theory) $\endgroup$
    – Emil Jeřábek
    Commented Aug 25, 2019 at 11:17
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    $\begingroup$ I don’t know what topological properties you are interested in, but as a trivial observation, since $S(A)$ is compact Hausdorff, its $G_\delta$ subspaces are Baire spaces. If moreover $S(A)$ is second-countable (i.e., when $A$ and the underlying first-order language are countable), then its $G_\delta$ subspaces are completely metrizable (i.e., Polish spaces). $\endgroup$
    – Emil Jeřábek
    Commented Aug 26, 2019 at 8:40
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    $\begingroup$ Also, the description in the comment doesn’t work out. Assuming you mean countable intersections and unions (otherwise it doesn’t make sense at all), if the complement of $\mathcal F$ is a countable union of countable intersections of open sets, that only makes $\mathcal F$ an $F_{\sigma\delta}$ set (i.e, $\Pi_3$ in the Borel hierarchy), not $G_\delta$. $\endgroup$
    – Emil Jeřábek
    Commented Aug 26, 2019 at 9:26

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