Prove that, if S is open, each of its points is a point of accumulation of S.
How can I prove this??
It's clearly untrue in arbitrary spaces, but since this is an advanced calculus book I'm going to take a wild guess and assume you meant and open subset of $\displaystyle \mathbb{R}^n$. If so then this is true if $\displaystyle S$ is non-empty. Let $\displaystyle s\in S$ and $\displaystyle U$ any neighborhood of it, then since $\displaystyle S$ is open we know there exists a neighborhood $\displaystyle N$ of $\displaystyle s$ such that $\displaystyle N\subseteq S$. Note then that $\displaystyle N\cap S$ is a neighborhood of $\displaystyle s$ contained in $\displaystyle s$. Choose some open ball $\displaystyle s\in B_{\delta}(s)\subseteq S\cap N$ and use the uncountability of open balls in euclidean space to find a point different from $\displaystyle s$ in $\displaystyle B_{\delta}(s)$.