Can some one please assist me with this problem
Let S=R^2\Q^2 (Points (x,y) in S have at least one irrational coordinate.) Is S connected? Prove or disprove.
The must be some point $\displaystyle C=\left( {\alpha ,\beta } \right) \in S$ that has both coordinates irrational.
Now suppose that $\displaystyle P=\left( {h,k} \right) \in S$ is any other point in $\displaystyle S$. Say that $\displaystyle h$ is irrational, at least one coordinate is.
Consider the three points $\displaystyle \left( {h,k} \right)\,,\,\left( {h,\beta } \right)\,\& \,\left( {\alpha ,\beta } \right)$.
Construct two line segments: $\displaystyle l_1 (t) = \left( {t\left( {\alpha - h} \right) + h,\beta } \right)\,\& \,l_2 (t) = \left( {h,t\left( {\beta - k} \right) + k} \right)\,;\,0 \le t \le 1$.
Every point on either line segment has one of its coordinates irrational. Thus both are subsets of $\displaystyle S$.
The union of those two line segments is connected by $\displaystyle \left( {h,\beta } \right)$, thus $\displaystyle P$ is connect to $\displaystyle C$ by way of the path.
Does this mean that $\displaystyle S$ is connected?