In his book Four pillars of geometry, John Stillwell defines informally (p. 107):
"[The real projective line] is the set $\mathbb{R} \cup \{\infty\}$ together with all the linear fractional functions mapping $\mathbb{R} \cup \{\infty\}$ onto itself."
Linear fractional functions are the functions $f(x) = \frac{ax + b}{cx +d}$ with $ad - bc \neq 0$. I wonder what together with does mean?
While the real line is just $\mathbb{R}$, it's clear what a point on the real line is: just an element $x \in \mathbb{R}$.
(Why don't we define the real line to be $\mathbb{R}$ together with the isometries $f(x) = \pm x + a$ - or do we implicitely do?)
Once again:
What does together with mean precisely and formally? Something like $\mathbb{R}\mathbb{P}^1 = \langle \mathbb{R} \cup \{\infty\}, LF\rangle$ with $LF = \{f(x) = (ax+b)/(cx +d)\ |\ a,b,c,d \in \mathbb{R}, ad - bc \neq 0\}$?
If $\mathbb{R}\mathbb{P}^1$ thus is an ordered pair of sets, what then is an element of resp. point on $\mathbb{R}\mathbb{P}^1$ considered as a set?
(Better might be $\mathbb{R}\mathbb{P}^1 = \mathbb{R} \cup \{\infty\} \times LF$, thus an element of $\mathbb{R}\mathbb{P}^1$ would be a pair $\langle x, f(\cdot)\rangle$ with $x \in \mathbb{R} \cup \{\infty\}$ and $f(\cdot) \in LF$.)
If it suffices to define the real line to be $\mathbb{R}$, why doesn't it suffice to define the real projective line to be $\mathbb{R} \cup \{\infty\}$?