I follow up to "There is a point $x^\prime\in \overline{E} \cap U_x$." But, I am struggling to understand the next sentence "Therefore every neighborhood of $x$ contains a point of $E$ and hence $x \in \overline{E}$".
Specifically, I agree that every neighborhood of $x$ contains a point of $E$. But, how does it lead to $x \in \overline{E}$?
I really appreciate if you elaborate on this.