Isolated point

Let $(X,d)$ be a metric space and let $A\subseteq X$. A point $x\in A$ is an isolated point of $A$ if there exists $r>0$ such that

Isolated points are the opposite of limit points: near an isolated point there are no other points of the set. Sets can have both isolated points and limit points.

Examples:

• In $\mathbb{R}$, every integer $n\in\mathbb{Z}$ is an isolated point of $\mathbb{Z}$ (take $r=1/2$).
• In $\mathbb{R}$, $1$ is an isolated point of $\{1\}\cup\{1/n:n\in\mathbb{N}\}$.
• In $\mathbb{R}$, no point of $(0,1)$ is isolated in $(0,1)$.