Let $S\subseteq \mathbb{R}$. A point $x\in S$ is an interior point of $S$ if there is an open interval $(a,b)$ for which $x\in (a,b) \subseteq S$. Denote the set of all interior points of $S$ by $\mathrm{int}(S)$.
- Prove that $S$ is an open set if and only if $S=\mathrm{int}(S)$.
- Prove that $\overline{S}$ and $\overline{\mathrm{int}(S)}$ need not be equal.
The definition of an open set:
A subset $U$ of a metric space $(M, d)$ is called open if, given any point $x \in U$, there exists a real number $\epsilon > 0$ such that, given any point $y \in M$ with $d(x, y) < \epsilon$, $y$ also belongs to $U$.
I have been thinking about 1. and 2. for 30 minutes and I am completely blank. Sorry.