Definition:Path-Connected/Topology/Set

Definition
Let $T = \struct {S, \tau}$ be a topological space.

Let $U \subseteq S$ be a subset of $S$.

Let $T' = \struct {U, \tau_U}$ be the subspace of $T$ induced by $U$.

Then $U$ is a path-connected set in $T$ every two points in $U$ are path-connected in $T\,'$.

That is, $U$ is a path-connected set in $T$ :
 * for every $x, y \in U$, there exists a continuous mapping $f: \closedint 0 1 \to U$ such that:
 * $\map f 0 = x$
 * and:
 * $\map f 1 = y$

Also known as
Some sources refer to this as a path-connected subset of $T$, but strictly speaking the subset nature of $U$ is of the underlying set $S$, not of $T$.

Thus, on, path-connected set is preferred, which is consistent with the concepts open set, closed set, connected set, and so on.