Let $ X$ be a variety with a $ G$ -action by an algebraic group on it.

My question refers to a motivating example from

https://web.maths.unsw.edu.au/~danielch/thesis/mbrassil.pdf

Here the relevant excerpt:

Here the author discusses an example of $ X/G$ in order to explaine that it is neccessary to form $ X/G$ as *categorical quotient* and not the *topological* one.

We consider following motivating example introduced at page 27:

Here we take $ X:= \mathbb{C}^2$ with action by $ G:=\mathbb{C}^x$ via multiplication $ \lambda \cdot (x,y) \mapsto (\lambda x, \lambda y)$ .

Obviously the “naive” topological quotient consists set theoretically of the lines $ \{(\lambda x, \lambda y) \vert \lambda \in \mathbb{C}^x \}$ and the origin $ \{(0,0)\}$ .

Topologically the origin lies in the closure of every line.

So the QUESTION is why does this argument already imply that $ Y:=X/G$ cannot have a strucure of a variety? I don’t understand the argument given by the author.

If we denote by $ p:\mathbb{C}^2 \to Y$ the canonical projection map and by (continuity?) this map can’t separate orbits, why does this already imply that $ Y$ doesn’t have structure of a variety as stated in the excerpt?

Especially which role does here play the fact that we can’t separate the lines from the origin (in *pure topologically way*)? Does it cause an obstacle in order to form a variety structure on $ X/G$ ?

Remark: I know that there are different ways to deduce that if we define $ X/G$ pure topologically then it cannot have a structure of a variety. The most common argument is to introduce the invariant ring $ R^G$ and to calculate it explicitely here. But the main issue of this question is it made me curious that the given argumentation seems to be a bit more “elementary” in sense that he doesn’t explicitely work in this example with the concept of the invariant ring $ R^G$ .