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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.

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    $\begingroup$ 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. $\endgroup$ Apr 8 at 9:22

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The quotient topology, by a linear flow of irrational slope, is the trivial topology on an uncountable set of the same cardinality as the reals.

When the slope is rational the quotient is much nicer; it is homeomorphic to $S^1$.

As a hint for the latter exercise: start with the case that the flow is horizontal and carefully produce a section (homeomorphic to the quotient).

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  • $\begingroup$ Thanks for your answer Sam - regarding the irrational case: I am not sure to see how the quotient topology on the quotient space be non-trivial (I can see why it's uncountable), which subsets of the quotient would have an open pre-image on T2? $\endgroup$
    – Hapax
    Apr 8 at 9:38
  • $\begingroup$ As far as I understand, the quotient space is topologically trivial when the slope is irrational. It should follow from this exercise: math.stackexchange.com/questions/2226618/… $\endgroup$ Apr 8 at 11:14
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    $\begingroup$ I assumed that “trivial” meant “trivial space”. I will rephrase. $\endgroup$
    – Sam Nead
    Apr 8 at 11:22
  • $\begingroup$ Thanks guys (and yes I was talking about the quotient topology indeed). $\endgroup$
    – Hapax
    Apr 8 at 12:09
  • $\begingroup$ 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. $\endgroup$
    – Lee Mosher
    Apr 8 at 18:25

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