Exercise 1, § 1

For every $\theta \in \mathbf{R}$, let $C_{\theta}$ be the segment of the plane with endpoints $(0,0)$ and (cos(2$\pi \theta$ ), sin(2$\pi \theta$ )); put

C = $\bigcup C_{\theta}$.

$\theta \in \mathbf{Q}\cap [1/6,1/4]$

(a) The space C is contractible at 0, but at no other point.

(b) Let D be the union of the translates $(0, n) + C$ of C, for $n\in \mathbf{Z}$. Prove that the space D is homotopic to a point but that it is not contractible at any of its points.