The must be some point that has both coordinates irrational.
Now suppose that is any other point in . Say that is irrational, at least one coordinate is.
Consider the three points .
Construct two line segments: .
Every point on either line segment has one of its coordinates irrational. Thus both are subsets of .
The union of those two line segments is connected by , thus is connect to by way of the path.
Does this mean that is connected?