I have the solution to the exercise but have a doubt on one thing, I state the exercise:
Given $$ X = \{ (x,y) \in R^2 | x = \frac{1}{n}, n \in N \}$$
and $Y = X/_{\sim}$ where the equivalence relation is $(x_1, y_1) \sim (x_2,y_2) \iff (x_1, y_1) = (x_2,y_2)$ or $ y_1 = y_2$ with $x_1,x_2 \in Q$
is it possible to give a homeomorphism from $Y$ to a metrizable space?
The start of the solution states that because $x_1 = 1/n$ and $x_2 = 1/m$ the product $x_1 x_2$ is always rational and this hints at the fact that the homeomorphism is going to be with a line ($R$).
My doubt is: why point out that $x_1,x_2 \in Q$ when we already know this and why would the product $x_1 x_2$ being in $Q$ hint at the fact the homeomorphism is going to be with a single line? could someone kindly explain this to me?
I don't really understand why we should care about the product $x_1x_2$, but a solution is as follows. Define $f:X\to \Bbb{R}:(x,y)\to y$. It is continuous and constant on $\simeq$, thus it factors through $\bar{f}:Y \to \Bbb{R}$. It is the just a matter of technique to show that $\bar{f}$ is a homeomorphism.