Show set is a polygon connected set?

[cookiesyum]

Homework Statement

Show that the complement of the set S = {all (x, y) with x and y rational numbers} is a polygon connected set. Is it an open set?

Homework Equations

The Attempt at a Solution

The complement of S = {all (x, y) with x not rational or y not rational}
Let the points (x, y) and (z, w) exist in the complement of S.

Case 1:

Suppose x is not rational and z is not rational.

Case 2:

Suppose x is not rational and w is not rational.

Case 3:

Suppose y is not rational and z is not rational.

Case 4:

Suppose y is not rational and w is not rational.

I don't really know how to actually prove my four cases.

 

[Billy Bob]

Example: (pi,2) to (e,4)

Line segment from (pi,2) to (pi,pi), and another line segment from (pi,pi) to (e,pi), and finally another segment from (e,pi) to (e,4).