Exercise 4, § 5

Soit P the subspace of $\mathbf{R}^2$ defined as in Exercise 5, I, p. 146, b). Let $a$ be the point $(0,0,1)$ of $\mathbf{R}^3$ and let C be the union of the segments $[a,(x,0)]$, for $x\in P$.

a) Prove that the space C is locally connected by arcs and simply connected by arcs.

b) Let D be the union of the space C and its symmetric $-C$. Prove that the space D is simply connected, but that it is not simply connected by arcs.

c) Prove that the Poincaré group $\pi_1(D,0)$ has the power of the continuum.