# Topology of windings on the two-torus

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.

• If the slope is irrational, a keyword to search for is 'irrational torus'. I know that there are interesting works regarding the diffeological structure of the quotient space. Apr 8 at 9:22

When the slope is rational the quotient is much nicer; it is homeomorphic to $$S^1$$.
• In the irrational case, since an equivalence class on $T^2$ is the same thing as an orbit of the flow, the question "Which subsets of the quotient would have an open pre-image on $T^2$?" is more-or-less the same as the question "Which open subsets $U \subset T^2$ are unions of flow orbits?". The answer is: only $\emptyset$ or $T^2$, because if $T^2-U \ne \emptyset$ then $U$ is one of those sets if and only if $T^2-U$ is a dense closed set. Apr 8 at 18:25