Definition:Path-Connected

Points in Topological Space
Let $T = \left({X, \vartheta}\right)$ be a topological space.

Let $a, b \in X$ be such that there exists a path from $a$ to $b$.

That is, there exists a continuous mapping $f: \left[{0. . 1}\right] \to X$ such that $f \left({0}\right) = a$ and $f \left({1}\right) = b$.

Then $a$ and $b$ are path-connected.

Topological Space
Let $T$ be a topological space.

Then $T$ is path-connected (or path connected) iff every two points in $T$ are path-connected 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 $f \left({1}\right) = y$.