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 $T$).

However, this terminology can obscure the notion that such an $H$ is specifically a subspace of $T$ induced by $\tau$.

If it is necessary for extreme clarity, the term connected subspace can be used.

Also see

 * Equivalence of Definitions of Connected Set