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It is well known that a complete metric space which has no isolated points is always uncountable.

Now let's remove the restriction that the metric space is complete. If the metric space has no isolated points, then I think it is NOT true that its always uncountable.

So I'm trying to come up with an example of a countable metric space with no isolated points.

Can somebody help me out with this? Is there an example of a metric space $(X,d)$ like this?

I thought about $X=\mathbb{Q}$ equipped with some weird distance function, like the discrete distance but that doesn't seem to work.

EDIT: My definition of an isolated point: $a\in X$ is isolated if there exists an $r>0$ such that $B(a;r)=\{a\}$. So if there are no isolated points in $X$, then for every $a \in X$, we have $B(a;r)\setminus \{a\} \neq \emptyset$ for all $r>0$.

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    $\begingroup$ Why would one need a weird distance function? $\endgroup$ – André Nicolas Nov 17 '15 at 6:25
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The metric space $\mathbb{Q}$ equipped with the Euclidean distance metric is countable, and has no isolated points.

We know it is countable; also, we know that every rational $x\in \mathbb{Q}$ has another rational arbitrarily close to it, so no $x\in\mathbb{Q}$ could be an isolated point.

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  • $\begingroup$ Okay I had a major brainfart and overlooked the fact that every point of $\mathbb{Q}$ is not isolated. For some strange reason I thought that they are all isolated. Thanks! $\endgroup$ – Greg.Paul Nov 17 '15 at 6:32

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