Exercise 6, § 1

Let X be the space $\mathbf{R}^2-\mathbf{Q}^2$.

a) Exercise prove that the space X is arcwise connected. b) Let $a, b$ be points of $\mathbf{R}-\mathbf{Q}$. Let $c_{a,b}:\mathbf{I}\rightarrow \mathbf{R}^2$ be the unique mapping which verifies $c(0) = (a, a),c(1/4) = (a, b),c(1/2) = (b, b),c(3/4) = (b, a)$ and $c(1) = (a, a)$ and whose restrictions to the intervals $[(m-1)/4, m/4]$ are affine, for $1\leqslant m\leqslant 4$. Exercise prove that $c_{a,b}$ is a loop in X.

c) Let $a, b, b'$ be elements of $\mathbf{R}-\mathbf{Q}$ such that $b=\not b'$. Exercise prove that the loops $c_{a,b}$ and $c_{a,b'}$ are not strictly homotopic.

d) Deduce that, for every $a\in \mathbf{R}-\mathbf{Q}$, the group $\pi_1(X,(a, a))$ has the power of the continuum.