Why is $(0,1)\cup \left\{ 3 \right\}$ disconnected?

In the subspace topology inherited from $\mathbb{R}$, the space $X = (0,1)\cup\{3\}$ can be covered by two open sets, $(0,1)$ and $\{3\}$, which are disjoint. Hence $X$ is disconnected.

It is also arcwise disconnected because no path can be found connecting any $x\in(0,1)$ with $3$.

Hint: Show that $\{3\}$ is a clopen set.