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?