Exercise 11, § 2

(a) Let $X$ be the quotient space of the space $\mathbf{R}\times \{1,2\}$ by the finest equivalence relation for which $(t,1)$ is equivalent to $(t,2)$ if $t\in \mathbf{R}^*$. Prove that there does not exist an injective path in X whose endpoints are the images of the points $(0,1)$ and $(0,2)$.

(b) Let $c:\mathbf{I}\rightarrow \mathbf{S}^1$ be the path defined by $c(t) =e^{3\pi it}$; it joins the point 1 to the point $-1$. Prove that there is no injective path in $\mathbf{S}_1$ which is strictly homotopic to the path $c$.