Definition:Connected (Topology)/Set/Definition 3

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 set of $T$ :
 * the topological subspace $\left({H, \tau_H}\right)$ of $T$ is a connected topological space.

Also see

 * Equivalence of Definitions of Connected Set