7
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$X=(-\infty,0]\cup\left\{{1\over n}:n\in\mathbb N\right\}$ with subspace topology. Then

  1. $0$ is an isolated point

  2. $(-2,0]$ is an open set

  3. $0$ is a limit point of the subset $\left\{{1\over n}:n\in\mathbb N\right\}$

  4. $(-2,0)$ is open set.

$0$ is not an isolated point of $X$ as every nbd of $0$ contains a point of the form $1\over n$ so $3$ is true and $1$ is false, $2$ is false as $0$ is not interior point, $4$ is true as every point is interior point. Am i right?

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  • 3
    $\begingroup$ You're right.${}$ $\endgroup$ – Ilya May 22 '13 at 10:46
  • $\begingroup$ only 3 and 4 are true. by the way; 3 is not a limit point. $\endgroup$ – user59671 May 22 '13 at 10:52
  • $\begingroup$ Then why $3$ is true? $\endgroup$ – miosaki May 22 '13 at 10:54
  • $\begingroup$ 3 is not a limit point. but 3 is true. because 0 is a limit point. $\endgroup$ – user59671 May 22 '13 at 20:54
3
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You are right.

3) is true as every nbd of 0 contains a point of the form $\frac 1n$.

4) is true as for any points $x\in (0,2)$, there exists an open set $U$ of $\mathbb R$ such that $x \in U \cap X \subseteq (0,2)$.

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