$\mathbb R/\mathbb Q$ is known to be compact, where topology on $\mathbb R$ is Euclid topology, and define $a~b$ is equivalent to $a-b\in\mathbb Q$, 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.
The continuous map $\mathbb R\to \mathbb R/\mathbb Q$ factors as $\mathbb R\to\mathbb R/\mathbb Z\to\mathbb R/\mathbb Q,.$
But $\mathbb R/\mathbb Z\cong S^1,$ which is compact, and the continuous image of a compact space is compact.
This is more general, if $X$ is a topological space with two equivalence relations $\sim_1,\sim_2$ such that $x\sim_1 y$ implies $x\sim_2 y,$ then if $X/\sim_1$ is compact, then $X/\sim_2$ is compact.