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Need to verify a few statement regarding metric spaces, found them in a few questions

Let $(X,d)$ be a metric space

1) $X= \bigcup_{n\in \Bbb N} B(a,n) $ where $B(a,r)$ indicates the open ball with centre a and radius r

2) if $ A \subset X $and $A$ is countable $\Rightarrow A^{o}=\emptyset $

3) $ \overline A \neq A^{o}$

$A \neq A^{o}$ indicates the set of interior points

My answers were that 1st was true and 2nd is wrong because in discrete metric any point in A can be an interior point, I think the 3rd is also false, are these correct

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  • $\begingroup$ If a set is both open and closed then $A=A^\circ=\overline A$; so $3$ is false in general. $\endgroup$ – Pedro Tamaroff Jul 2 '13 at 18:13
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Yes, you are correct. For the third one, take $A$ as the emptyset...

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