I am trying to prove the following result: Fix [itex]a,b \in \mathbb{R}[/itex] with [itex]a \neq 0[/itex]. Let [itex]L = \{(x,y) \in \mathbb{R}^2:ax+by = 0\}[/itex] and let [itex]\pi:\mathbb{R}^2 \rightarrow \mathbb{T}^2[/itex] be the canonical projection map. If [itex]\frac{b}{a} \notin \mathbb{Q}[/itex], then [itex]\pi(L)[/itex] (with the subspace topology) is not a submanifold of [itex]\mathbb{T}^2[/itex].(adsbygoogle = window.adsbygoogle || []).push({});

I am having difficulty however showing that [itex]\pi(L)[/itex] is not locally Euclidean. From drawing a few pictures, I think every neighborhood of [itex]\pi(0)[/itex] is disconnected (which would be enough to complete the proof), but I am having difficulty showing this. Any help?

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# Irrational Winding of the Torus

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