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

Question

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.

Can you clarify exactly what you mean by "indiscrete metric space"? — Anthony Carapetis, May 07 '17 at 05:16
Presumably it means aside from the discrete metric, meaning distance between any two points is 1 (or some other number). — Amin Idelhaj, May 07 '17 at 05:22
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. — Anthony Carapetis, May 07 '17 at 05:24
If $X$ is a non-empty finite set then any metric on $X$ is topologically equivalent to the discrete metric on $X$.So when we are talking about an indiscrete metric space we must have thought about metrices on an infinite set $X$.Isn't it? — Arnab Chattopadhyay., May 07 '17 at 05:36

score 2 · Accepted Answer · answered May 07 '17 at 17:12

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.

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

1 Answers1