Definition:Disconnected (Topology)/Set/Definition 2

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

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

$H$ is a disconnected set of $T$ there exist open sets $U$ and $V$ of $T$ such that all of the following hold:
 * $H \subseteq U \cup V$
 * $H \cap U \cap V = \O$
 * $U \cap H \ne \O$
 * $V \cap H \ne \O$

Also see

 * Equivalence of Definitions of Disconnected Set