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$X=(-\infty,0]\cup\left\{{1\over n}:n\in\mathbb N\right\}$ with subspace topology. Then
$0$ is an isolated point
$(-2,0]$ is an open set
$0$ is a limit point of the subset $\left\{{1\over n}:n\in\mathbb N\right\}$
$(-2,0)$ 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 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 $x \in U \cap X \subseteq (0,2)$.
