Exercise 2.8

Open sets: Yes. Any point $x$ in an open set $E$ is contained in a neighborhood $N_{𝜖}(x)\subset E$. Any point $y$ such that $d(y,x)<𝜖$ is contained in $E$. It is clear that any neighborhood of $x$ contains such a point $y$.

Closed sets: No. A single point set $E=\{x\}$ is closed, but is not a limit point, since no neighborhood of $x$ contains a point $y\in E$ such that $y\ne x$.