Proof:

The proof is broken into several steps. We shall employ the exponential map ex: $\mathbb{R} \longrightarrow S^1$ given by

   ex$$(t)= e^{2\pi i t}. \eqno(12.1)$$

The function ex maps $(\frac{-1}{2},\; \frac{1}{2})$ homeomorphically onto $S^1 -\{-1\}$ and we denote its inverse by

   lg$$: S^1 -\{ -1 \} \longrightarrow ( \frac{-1}{2},\; \frac{1}{2}) \eqno(12.2)$$

which is also a homeomorphism.