Definition:Isolated Point (Topology)

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

Isolated Point of Subset
Let $H \subseteq S$ be any subset of $S$.

Then $x \in H$ is an isolated point of $H$ iff:
 * $\exists U \in \tau: U \cap H = \left\{{x}\right\}$

That is, iff there exists an open set of $T$ containing no points of $H$ other than $x$.

Isolated Point of Space
When $H = S$ the definition applies to the entire topological space $T = \left({S, \tau}\right)$:

$x \in S$ is an isolated point of $T$ iff:
 * $\exists U \in \tau: U = \left\{{x}\right\}$

That is, iff there exists an open set of $T$ containing no points of $S$ other than $x$.