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.

  • $\begingroup$ What are $F_\sigma$ and $G_\delta$? $\endgroup$ – Joppy May 7 '17 at 5:08
  • 1
    $\begingroup$ @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

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.