$mathbb R/mathbb Q$ is known to be compact, where topology on $mathbb R$ is Euclid topology, and define $a~b$ is equivalent to $a-binmathbb Q$, topology on $mathbb R/mathbb Q$ is given by quotient topology.

Then, I want to prove $mathbb R/mathbb Q$ is compact.

I know in general, ‘Let $G$ be a topology group. and $H$ is dense in $G$, then, $G/H$ is trivial topology’.

But I want to prove the titled statement without using the fact above.