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Which of the following statements are correct?

The set $\{0\} \cup \{\frac{1}{n} |n \in \mathbb{N}\}$ is compact in $\mathbb{R}$.

$\mathbb{N}$, the set of all natural numbers, is both closed and open in $\mathbb{R}$.

$\{\frac{1}{n} |n \in \mathbb{N}\}$ is compact in $\mathbb{R}$.