Definition:Path-Connected

Definition
Let $T$ be a topological space.

Then $T$ is path-connected (or path connected) iff every two points in $T$ can be joined by a path in $T$.

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