Equivalence of Definitions of Totally Pathwise Disconnected Space

Proof
Let $T = \left({S, \tau}\right)$ be a totally pathwise disconnected space by definition 1.

That is:
 * all path components of $T$ are singletons.

By definition of path component, this means there are no two points $x, y \in S$ such that there exists a path between $x$ and $y$.

By definition of paths, this means the only paths in $T$ are from a single point to that single point.

That is, that the paths in $T$ are constant mappings.

By definition of path:
 * the only continuous mappings from the closed unit interval $\left[{0 \,.\,.\, 1}\right]$ to $T$ are constant mappings.

That is the definition of a totally pathwise disconnected space by definition 2.

Let $T = \left({S, \tau}\right)$ be a totally pathwise disconnected space by definition 2.

That is:
 * the only continuous mappings from the closed unit interval $\left[{0 \,.\,.\, 1}\right]$ to $T$ are constant mappings.

That is, that the paths in $T$ are constant mappings.

That is, the path components are singletons.

That is the definition of a totally pathwise disconnected space by definition 1.