isolated point

Isolated point

Let $D\subseteq \mathbb{R}$. A point $x_0\in D$ is isolated if some epsilon neighbourhood contains no other point of $D$:

$$\exists \epsilon >0:U_{\epsilon }(x_0)\cap D=\{x_0\}$$

A gap separates $x_0$ from the rest of $D$. Every point of $D$ is either isolated or a limit point of $D$.

Example

Each $1/n$ in $\{1/n:n\in \mathbb{N}\}$ is isolated (see the figure in limit point).
Every integer in $\mathbb{Z}$ is isolated, its nearest neighbours sit a distance $1$ away.
No point of an interval like $(0,1)$, nor of $\mathbb{Q}$, is isolated: every neighbourhood is crowded.

A set all of whose points are isolated is called discrete.