Consider the componentwise complex conjugation $\tau': \mathbb{C}^{n+1}\setminus\{0\}\rightarrow \mathbb{C}^{n+1}\setminus\{0\}$.
I have to show that this map "induces" a map $\tau$ on the complex projective space $\mathbb{C}P^n$ (and furthermore that the set of fixed points of $\tau$ is homeomorphic to $\mathbb{R}P^n$, but at the moment I only care about the first point).
I interpret "inducing" in the sense that I have to show that $\tau'$ is constant on equivalence classes. However trying to do this I get $\tau'(\lambda c) = \overline{\lambda} \tau'(c) (\neq \tau'(c)$ in general), where $\lambda \in \mathbb{C} \setminus \{0\}$ and $c \in\mathbb{C}^{n+1}\setminus\{0\}$.
Where is the mistake in my thoughts?