Equivalence of Definitions of Totally Pathwise Disconnected Space

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

Then $T$ is totally pathwise disconnected the only continuous mappings from the closed unit interval $\left[{0 \,.\,.\, 1}\right]$ to $T$ are constant mappings.

Proof
By definition of paths, the statement can be worded as:

Then $T$ is totally pathwise disconnected the only paths in $T$ are from a single point to that single point.

That is, the path components are singletons.

This is the definition of a totally pathwise disconnected space.