Exercise 1, § 1

Let S be the graph of the mapping from $]0,1[$ into $\mathbf{R}$ given by $t\mapsto$ sin($\pi /t$). Let W be the union of S and the image of a path in $\mathbf{R}^2-$ S joining the points $(0,0)$ and $(1,0)$ (“Warsaw circle”).

(a) Prove that the space W is simply connected by arcs but is not simply connected.

(b) For every integer $n\geqslant 0$, put $X_n=\mathbf{R}/2^n\mathbf{Z}$ and denote by $f_n: X_{n+1}\rightarrow X_n$ the canonical surjection; let X be the space $\varprojlim_n X_n$. Prove that the space X is simply connected by arcs but is not simply connected.