Existence of Connected Space which is Totally Pathwise Disconnected

Theorem
There exists at least one example of a topological space which is both connected and totally pathwise disconnected.

Proof
Let $T$ be Gustin's sequence space.

From Gustin's Sequence Space is Connected, $T$ is a connected space.

From Gustin's Sequence Space is Totally Pathwise Disconnected, $T$ is a totally pathwise disconnected space.

Hence the result.