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I can think of a countable topological space with a finite number of limit points. I was wondering if the set of limit points could also be infinite.

To make it more clear, I'm asking if there is a countable topological space $X$ with a infinite subset $A$ such that every $x\in A$ is a limit point of $X$.

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    $\begingroup$ Well there is the lame answer that all points in a given set are limit points of the set, so you can just take any countable set in whatever space you want. But you can also do something like taking the rational numbers and removing the integers or something. This way the integers are all limit points. $\endgroup$ – Dionel Jaime Dec 29 '18 at 8:36
  • $\begingroup$ @Dionel Jaime Yes I was thinking of something along those lines, I'll edit the post to restrict the question to metric spaces. $\endgroup$ – Markus Steiner Dec 29 '18 at 8:38
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    $\begingroup$ This doesn't change anything. The rationals are still a metric space under the (induced) subspace topology. $\endgroup$ – Dionel Jaime Dec 29 '18 at 8:39
  • $\begingroup$ @Dionel Jaime You are right, let's just say a subspace of R? $\endgroup$ – Markus Steiner Dec 29 '18 at 8:43
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    $\begingroup$ Rational numbers are also real numbers my friend ... $\endgroup$ – Dionel Jaime Dec 29 '18 at 8:45
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Take Q, the space of rational numbers. Every point is a limit point. So there's a countable set of limit points.

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