Definition:Connected (Topology)/Set/Definition 2

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

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

Then $H$ is disconnected iff there exist open sets $U$ and $V$ in $S$ 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 iff it is not disconnected.