Homogeneous Coordinates on a Real Projective Space

Question

I attempt to understand the construction of the standard $C^{\infty}$ atlas on a real projective space from Loring Tu's An Introduction to Manifolds (Second Edition, page no. 79). Tu denotes the equivalence class of a point $(a^0, \cdots, a^n) \in \mathbb{R}^{n+1} - \lbrace 0 \rbrace$ by $[a^0, \cdots, a^n]$ and define $\pi: \mathbb{R}^{n+1} - \lbrace 0 \rbrace \to \mathbb{R}P^n$ be the projection. He calls $[a^0, \cdots, a^n]$ the homogeneous coordinates on $\mathbb{R}P^n$ (page no. 76).

If I understand correctly, then $a^0, \cdots, a^n$ in $(a_0, \cdots, a^n)$ are all real numbers. But then on page no. 79, he writes,

My Questions

1. How is $a_0$ is even considered to be a function on $\mathbb{R}P^n$?
2. What does this sentence mean?

Although $a^0$ is not a well-defined function on $\mathbb{R}P^n$, the condition $a^0 \neq 0$ is independent of the choice of a representative for $[a^0 , \cdots, a^n]$.

Answer 1

You can multiply all the homogeneous coordinates by a common factor of $\lambda\ne 0$. Clearly, $a^0 \ne 0$ if and only if $\lambda a^0\ne 0$.

Answer 2

In projective space, for example, $[1: 2: 3] = [2: 4: 6]$, so it doesn't make sense to say "what's the first coordinate of this point?" However, it does make sense to ask "is the first coordinate of this point zero?"