\begin{definition}[Definition:Path-Connected/Topology/Set]
Let $T = \left({S, \tau}\right)$ be a topological space.
Let $U \subseteq S$ be a subset of $S$.
Let $T' = \left({U, \tau_U}\right)$ be the subspace of $T$ induced by $U$.
Then $U$ is a '''path-connected set in $T$''' {{iff}} every two points in $U$ are path-connected in $T\,'$.
That is, $U$ is a '''path-connected set in $T$''' {{iff}}:
:for every $x, y \in U$, there exists a continuous mapping $f: \left[{0 \,.\,.\, 1}\right] \to U$ such that $f \left({0}\right) = x$ and $f \left({1}\right) = y$.
\end{definition}