# Is topology on $\mathbb{R}/\mathbb{Z}$ compact?

Is topology on $\mathbb{R}/\mathbb{Z}$ compact? If it is, how to prove it?

$\mathbb{R}/\mathbb{Z}$ denotes the set of equivalence classes of the set of real numbers, two real numbers being equivalent if and only if their difference is an integer.

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## 2 Answers

$\bf Hint:$ Find a compact subset $K$ of $\mathbb R$ so that the quotient map $\pi: \mathbb R\to \mathbb {R/Z}$ restricted to $K$ is onto.

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Or use that $S^1$ is compact and try to write down a homeomorphism between $\mathbb R/ \mathbb Z$ and $S^1$.

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