In short my question is: what can we say about the quotient topology induced by the linear flow on a two-torus? I know that an irrational slope leads to a dense winding and hence (if I'm not mistaken) the quotient topology is trivial. But what about the case where the slope is rational? I know orbits are then homeomorphic to $S^1$ but I can't get my head around what the topology of the quotient space looks like.

Thank you.