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In any metric space prove that every open set is $G_{\delta}$ set and every closed set is $F_{\sigma}$ set.(Hint: use the continuity of $x\longmapsto d(x,A)$.)

I tried to prove this by saying: If $U$ is a open set, consider $\bigcup_{i=0}^{\infty} A_{n}$, where $A_{n}=\bigcup_{x\in U} B(x,1+1/n)$ and $A_{0}=U$.

Hence every Open set is $G_{\delta}$ set.

Also I didn't get how to use the hint to solve the problem. Thanks in advance.

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    Did you switch the two notions? [Wikipedia](http://en.wikipedia.org/wiki/F%CF%83_set): In metrizable spaces, every open set is an $F_\sigma$ set. The complement of an $F_\sigma$ set is a $G_\delta$ set. In a metrizable space, any closed set is a $G_\delta$ set.2012-10-03
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    See also this question: [Every open set in $\mathbb R$ is $F_\sigma$ set](http://math.stackexchange.com/questions/196027/every-open-set-in-mathbbr-is-f-sigma-sets)2012-10-03
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    This is a home work question.Can any one say how to use the hint to solve the question.2012-10-03

2 Answers 2