\begin{definition}[Definition:Path-Connected/Topology/Topological Space]
Let $T = \struct {S, \tau}$ be a topological space.
Then $T$ is a '''path-connected space''' {{iff}} $S$ is a path-connected set of $T$.
That is, $T$ is a '''path-connected space''' {{iff}}:
:for every $x, y \in S$, there exists a continuous mapping $f: \closedint 0 1 \to S$ such that $\map f 0 = x$ and $\map f 1 = y$.
\end{definition}