Definition:Path-Connected/Topology/Topological Space

Definition
Let $T = \struct {S, \tau}$ be a topological space.

Then $T$ is a path-connected space $S$ is a path-connected set of $T$.

That is, $T$ is a path-connected space :
 * for every $x, y \in S$, there exists a continuous mapping $f: \closedint 0 1 \to S$ such that:
 * $\map f 0 = x$
 * and:
 * $\map f 1 = y$

Also known as
Some sources do not hyphenate, but instead report this as path connected.

Also see

 * Definition:Locally Path-Connected Space