Are the following statements true or false? Give brief justifications for your answers.

(i) If $X$ is a connected open subset of $\mathbb{R}^{n}$ for some $n$, then $X$ is path-connected.

(ii) A cartesian product of two connected spaces is connected.

(iii) If $X$ is a Hausdorff space and the only connected subsets of $X$ are singletons $\{x\}$, then $X$ is discrete.