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I need to find 3 countable infinite set in R such that a) every point of R is an accumulation point b) The set has infinitely many accumulation points, none of which are in the set c) the set has infinitely many accumulation points all of which are in the set

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  • $\begingroup$ There isn't any set that respects all three conditions. $\endgroup$ – Git Gud Oct 19 '13 at 22:59
  • $\begingroup$ @GitGud Not true. There is an obvious countable set that satisfies all three conditions. Condition b) is phrased awkwardly. What is meant is "there are infinitely many points not in the set that are accumulation points of the set". $\endgroup$ – Andrés E. Caicedo Oct 19 '13 at 23:07
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    $\begingroup$ @GitGud: As I read the question, it’s calling for three different sets, one satisfying (a), one satisfying (b), and one satisfying (c). $\endgroup$ – Brian M. Scott Oct 19 '13 at 23:09
  • $\begingroup$ @Andrés: I think that (b) means exactly what it says: the set has infinitely many accumulation points, and they’re all in the complement of the set. $\endgroup$ – Brian M. Scott Oct 19 '13 at 23:10
  • $\begingroup$ @AndresCaicedo In addition to concurring with BrianM.Scott, I read (a),(b) and (b),(c) as mutually exclusive. $\endgroup$ – Jonathan Y. Oct 19 '13 at 23:13
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HINT: There is a really obvious answer to (a). For (b) and (c) you could start by thinking about translates of the set $\left\{\frac1n:n\in\Bbb Z^+\right\}$.

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