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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 ?

  1. $S$ is infinite.
  2. $S$ contains a non-zero rational number.
  3. $S$ is connected.
  4. $S$ is a closed subset of $\mathbb{R}$
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