# Math Help - Topology and Analysis

1. ## Topology and Analysis

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.

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