Let $(X, d)$ be a path connected metric space with at least two elements, and let $S=\left\{d(x, y):x, y \in X\right\}$. Which of the following statements is not necessarily true ?
- $S$ is infinite.
- $S$ contains a non-zero rational number.
- $S$ is connected.
- $S$ is a closed subset of $\mathbb{R}$