# Definition:Isolated Point (Topology)/Subset

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## Definition

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

Let $H \subseteq S$ be a subset of $S$.

### Definition 1

$x \in H$ 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 $x$.

### Definition 2

$x \in H$ is an **isolated point of $H$** if and only if $x$ is not a limit point of $H$.

That is, if and only if $x$ is not in the derived set of $H$.

## Also see

- Results about
**isolated points**can be found**here**.