Connectedness of $\{(x,y)| x/y \in \mathbb Q\} \cup \{(x,0)|x\ne 0\}$

Question

I am trying to decide if the set $\{(x,y)| x/y \in \mathbb Q\} \cup \{(x,0)|x\ne 0\}$ is connected or disconnected. It is clearly not path-connected because it is impossible to get from line to the other without going through the origin, which is missing. I think it is also disconnected because it can be partitioned into two open sets by the line $y = \sqrt{2} x$. Is that right?

Answer 1

Yes, you are right.

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Alternatively, the set does not contain the origin, hence we have a continuous projection $(x,y)\mapsto \frac1{sqrt{x^2+y^2}}(x,y)$, which has a countable (hence disconnected) subset of $S^1$ as image.