Definition:Connected (Topology)/Set/Definition 2

Definition
Let $T = \left({S, \tau}\right)$ be a topological space.

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

$H$ is disconnected in $T$ there exist open sets $U$ and $V$ of $T$ such that:
 * $H \subseteq U \cup V$
 * $H \cap U \cap V = \varnothing$
 * $U \cap H \ne \varnothing$

and:
 * $V \cap H \ne \varnothing$

$H$ is connected in $T$ it is not disconnected in $T$.

Also see

 * Equivalence of Definitions of Connected Subset