Definition:Connected (Topology)/Set/Definition 1

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

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

$H$ is a connected subset of $T$ it is not the union of any two non-empty separated sets of $T$.

Also see

 * Equivalence of Definitions of Connected Subset