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Does there exist any non discrete metric space $(X,d)$ in which every $F_{\sigma}$ (resp. $G_{\delta}$) set is clopen?

I can't find any non discrete metric space $(X,d)$ having the above mentioned property.Please help me in finding this (if any).

EDIT $:$

If $X$ is countable then each of it's subset being countable can be expressed as a countable union of singleton sets each of which is closed.So every subset of $X$ is $F_{\sigma}$ and consequently open by the given condition i.e. $(X,d)$ becomes a discrete metric space.

If $X$ is uncountable then as above every countable subset of $X$ is open.Also if one have any co-countable set then it is obviously closed and hence $F_{\sigma}$. Cosequently by the given condition it is open.I find difficulty to prove the result for any other uncountable subsets of $X$.

Please help me in proving it (if it is possible).

Thank you in advance.

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  • $\begingroup$ What are $F_\sigma$ and $G_\delta$? $\endgroup$ – Joppy May 7 '17 at 5:08
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    $\begingroup$ @Joppy: en.wikipedia.org/wiki/G%CE%B4_set $\endgroup$ – Anthony Carapetis May 7 '17 at 5:15
  • $\begingroup$ Can you clarify exactly what you mean by "indiscrete metric space"? $\endgroup$ – Anthony Carapetis May 7 '17 at 5:16
  • $\begingroup$ Presumably it means aside from the discrete metric, meaning distance between any two points is 1 (or some other number). $\endgroup$ – Daminark May 7 '17 at 5:22
  • $\begingroup$ I would assume it's the stronger claim that the generated topology is not discrete; i.e. there exists a non-open set. Otherwise there are easy finite examples. $\endgroup$ – Anthony Carapetis May 7 '17 at 5:24
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No, there is no such space. In a metric space, every one-point set is both an $F_\sigma$ and a $G_\delta.$ If every $F_\sigma$ (or every $G_\delta$) is open, then every one-point set is open, so the space is discrete.

In any topological space, the following are equivalent:
(1) every $F_\sigma$ is clopen;
(2) every $G_\delta$ is clopen; (3) every closed set is open;
(4) every open set is closed.

The only $\text{T}_0$ spaces with these properties are the discrete spaces.

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