TITLE: Wikipedia's definition of isolated point.
QUESTION [8 upvotes]: 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.

REPLY [6 votes]: It should (and now does) say that $x$ is isolated in $S$ iff $\{x\}$ is open in $S$, not in $X$.

REPLY [3 votes]: This is true only if $S$ is open. Notice that in your example, $S=\{1\} \cup (3, 4)$ is not an open set in the standard topology on $\mathbb{R}$.