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

$X=(-\infty,0]\cup\left\{{1\over n}:n\in\mathbb N\right\}$ 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?

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

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 $x \in U \cap X \subseteq (0,2)$.