# Definition:Disconnected (Topology)/Set

## Definition

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

Let $H \subseteq S$ be a non-empty subset of $S$.

### Definition 1

$H$ is a disconnected set of $T$ if and only if it is not a connected set of $T$.

### Definition 2

$H$ is a disconnected set of $T$ if and only if there exist open sets $U$ and $V$ of $T$ such that:

$H \subseteq U \cup V$
$H \cap U \cap V = \O$
$U \cap H \ne \O$

and:

$V \cap H \ne \O$