Exercise 1, § 6

Let X be a topological space. The suspension of X is called the quotient of the space $X\times [0,1]$ by the finest equivalence relation for which $X\times \{0\}$ and $X\times \{1\}$ are equivalence classes; this space is denoted by S(X) and $p: X\times [0,1]\rightarrow S(X)$ is the canonical surjection.

(a) Prove that, for every integer $n\geqslant 0$, the space $S(\mathbf{S}_n)$ is homeomorphic to $\mathbf{S}_{n+1}$.

(b) Prove that the space S(X) is simply connected if and only if the space X is connected.