Isolated point

Definition

Let $X$ be a topological space, let $S \subseteq X$ and let $x \in S$. The point $x$ is an isolated point of $S$ when there is an open set $U$ of $X$ with $U \cap S = \{x\}$.

Remarks

Each point of $S$ is either an isolated point of $S$ or a limit point of $S$, and not both. In $\mathbb{R}$, each point of $\mathbb{Z}$ is an isolated point of $\mathbb{Z}$: take $U = (n - 1/2, n + 1/2)$. A point $x$ is an isolated point of $X$ itself if and only if $\{x\}$ is open.

Explanations

L1 human

An isolated point is a point of the set that stands alone: a small region about it contains no other point of the set. In the set $\{0\} \cup [1, 2]$, the point $0$ is isolated, but $1$ is not.