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 $x$.

Also see

• Results about isolated points can be found here.

Sources

• Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (1970)... (previous)... (next): $\text{I}: \ \S 1$: Limit Points