Definition:Connected (Topology)/Set

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

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

Also known as
A connected set $H$ of a topological space $T = \left({S, \tau}\right)$ is often found referred to as a connected subset (of $T$).

Some sources refer to the concept of a connected subspace, which is no more than a connected set under the subspace topology.

On, if the distinction is required, it will be specified explicitly.

Also see

 * Equivalence of Definitions of Connected Set