Isolated Point

Definition

Let $(X,d)$ be a metric space and $A\subseteq X$. A point $a\in A$ is an isolated point of $A$ if there exists $\epsilon >0$ such that: $B(a,\epsilon )\cap A=\{a\}$

Properties

1. An isolated point is an adherent point that is not a limit point.
2. If every point in $A$ is an isolated point, $A$ is called a discrete set.
3. In the subspace topology, $a$ is an isolated point iff the singleton set $\{a\}$ is open in $A$.