Wikipedia defines an isolated point of a subset $S \subseteq X$ to be a point $x \in S$ such that there exists a neighborhood $U$ of $x$ not containing any other points of $S$. Furthermore, it claims that this is equivalent to saying $\{x\}$ is open in $X$.
Question: How is the last sentence true? This seems to be false since for example $1$ is an isolated point of $\{1\} \cup (3, 4)$, but $\{1\} \subseteq \mathbb{R}$ is not open.
Best Answer
It should (and now does) say that $x$ is isolated in $S$ iff $\{x\}$ is open in $S$, not in $X$.