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Consider $\mathbb{R}$ equipped with a range of different topologies: discrete; indistcrete, Euclidean, cofinite, cocountable and lower limit. Suppose we have a sequence of real numbers given by $(x_n)_{n\in N} = (\frac{1}{n})_{n\in N}$. I'm trying to solve these questions on convergence in these different topologies: Does this sequence converge? If yes, to what points in $\mathbb{R}$?

I've found the following solutions:

  • Discrete: does not converge;
  • Indiscrete: convergence to all $x \in \mathbb{R}$;
  • Euclidean: converges to $0$;
  • Cofinite: convergence to all $x \in \mathbb{R}$;
  • Cocountable: does not converge;
  • Lower limit: converges to $0$.

Are these results correct? I am not looking for a full proof or explanation in each case, as I think I am capable writing it out. Just wondering if my rough intuition is correct :)

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    $\begingroup$ All the answers seem correct to me. :) $\endgroup$
    – Sangchul Lee
    Commented Jun 5, 2022 at 12:34
  • $\begingroup$ @SangchulLee great! thanks! $\endgroup$
    – Geigercounter
    Commented Jun 5, 2022 at 12:39
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    $\begingroup$ I’m voting to close this question because it was answered in the comments. $\endgroup$
    – José Carlos Santos
    Commented Jun 12, 2022 at 9:04

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