Totally Disconnected Space is Totally Pathwise Disconnected
Then $T$ is a totally pathwise disconnected space.
Then by definition $T$ contains no non-degenerate connected sets.
That is, there exist two points $x, y \in S$ such that there exists a path between $x$ and $y$.
That is, $x$ and $y$ are path-connected.
Thus they are in the same path component.