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Connected Set Topology on R^2 \ Q^2

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Let S = R^2 Q^2. Points (x,y) in S have at least one irrational coordinate.

Is S connected? Can we disprove with a counterexample?

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Connected Set Topology on R^2 Q^2 is investigated. The solution is detailed and well presented. The response was given a rating of "5/5" by the student who originally posted the question.

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The answer is YES.
Proof:
We want to show that S is connected. This means for any two points A=(x1,y1) and B=(x2,y2) in S, we can find a
path p in S, connecting A and B.
For the point A=(x1,y1), without the loss of generality, we can assume that x1 is ...

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