Definition:Totally Pathwise Disconnected Space/Definition 2

Definition
A topological space $T = \left({S, \tau}\right)$ is totally pathwise disconnected the only continuous mappings from the closed unit interval $\left[{0 \,.\,.\, 1}\right]$ to $T$ are constant mappings.

Also see

 * Equivalence of Definitions of Totally Pathwise Disconnected Space