Definition:Path-Connected/Topology/Topological Space

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

Then $T$ is path-connected iff $S$ is path-connected in $T$.

That is, $T$ is path-connected iff:
 * for every $x, y \in S$, there exists a continuous mapping $f: \left[{0 \,.\,.\, 1}\right] \to S$ such that $f \left({0}\right) = x$ and $f \left({1}\right) = y$.