# Definition:Isolated Point (Topology)

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## 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.