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Show $\mathbb{R}^2 - \mathbb{Q}^2$ is connected.

Now, I feel like showing that this is path connected is pretty simple, but showing that it is connected? The definition the we learned in class of a space being connected is:

A space $X$ is connected if and only if for all continuous maps $\alpha \rightarrow {0,1}$ we have that $\alpha$ is not onto.

I think another equivalent definition of connectedness may be more appropriate for this problem.

Anyway, can somebody help me out with this problem? Thanks!

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    If it is path-connected then it is connected. – Grešnik Jun 18 '19 at 21:05
  • This has been solved many times here (even without showing path-connectedness), e.g. https://math.stackexchange.com/questions/2300781/prove-that-mathbbr2-setminus-mathbbq2-is-connected-without-path-connect – Con Jun 18 '19 at 21:06

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