Definition:Isolated Point (Topology)

Definition

Let $T = \struct {S, \tau}$ be a topological space.

Isolated Point of Subset

$x \in H$ is an isolated point of $H$ if and only if:

$\exists U \in \tau: U \cap H = \set x$

That is, if and only if 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 $T = \struct {S, \tau}$:

$x \in S$ is an isolated point of $T$ if and only if:

$\exists U \in \tau: U = \set x$

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

The set of isolated points of $H \subseteq S$ can be denoted $H^i$.

Also see

• Results about isolated points can be found here.