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I would like to check whether the quotient space is compact or not. I know compact space's quotient space is also compact because projection map is continuous. But, what about the quotient space which is not compact?

I would like to formally prove the following cases ($\mathbb{R}$'s topology is usual Euclid topology here).

$\mathbb{R}/\mathbb{Q}$ is compact

Thank you in advance.

Martin Sleziak
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Pont
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1 Answers1

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There is a general result asserting that if $G$ is topological group and $H$ is a dense subgroup, then $G/H$ has the trivial topology, i.e. the only open sets are the empty set and $G/H$ itself. Thus, $\mathbb{R}/\mathbb{Q}$ has a trivial topology. In particular, it is compact.

For a proof of the above general result, see here.

uniquesolution
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