Question in Example 2.6.4 Page 57 S. G. Krantz Topology

Question

This example is from S. G. Krantz ( A guide to Topology) and I have a question in an example on page 57 in section: Quotient Topology.

I am unable to understand why this map is 2-to-1 and why 2 points are mapped to same point under π iff a=-b.

Answer 1

That $\pi : S^N \to \mathbb P^N$ is $2$-to-$1$ means that there are exactly two points $a,b \in S^N$ such that $\pi(a) = \pi(b)$. Notice that the last equation tells us that $b = \lambda a$ for some non-zero real number $\lambda$, but $$1=\|b\| = \|\lambda a\| = |\lambda|\|a\| = |\lambda|,$$ so $\lambda = -1$ since $a \neq b$. Of couse, if $a=-b$, then $a \sim b$ and $\pi(a) = \pi(b)$.