Connectedness: Is R^2 \ Q^2 connected or disconnected?

[passionate]

I have to show whether R^2\Q^2 is connected or disconnected. I think that it's connected since Q^2 is countable and R^2\ X where X is a countable set, then R^2\X is connected, but I haven't learned this fact yet. Anyone can help me with this problem please.

 

[daon]

I have to show whether R^2\Q^2 is connected or disconnected. I think that it's connected since Q^2 is countable and R^2\ X where X is a countable set, then R^2\X is connected, but I haven't learned this fact yet. Anyone can help me with this problem please.

Take any point x in X=R^2\Q^2. Then x=(x1,x2) and must satisfy that either x1 is irrational or x2 is irrational. Define y=(y1,y2) similarly. For a complete proof, one should consider all cases, but I'll show the case where x1 is irrational and y2 is irrational. Create a path from [0,1] to your set by p(t)=(x1, x2+[y2-x2]t). Note this entire line lies within your set X and connects (x1,x2) to (x1,y2). Note y2 is irrational. We can create a second path q(t)=(x1+(y1-x1)t,y2) that connects (x1,y2) with (y1,y2) and also lies strictly within X.

The resulting path (p*q)(t) is a path from (x1,x2) to (y1,y2) [* is concatenation]. Show the other cases and note path connectedness implies connectedness.

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