Consider the space of integer points $\Bbb{Z}^2=\{(x,y)|x,y\in\Bbb{Z}\}$.
Consider now the equivalence relation: $$ (x,y) \sim (x',y') \quad \Leftrightarrow \quad \beta x'=\alpha x,\, \beta y'=\alpha y \;\mbox{ for some } \alpha,\beta\in\Bbb{Z}\backslash\{0\}. $$
What is the space $\big(\Bbb{Z}^2\backslash\{(0,0)\}\big)/\sim$ ? Does it have a name? Is it related to the projective line (for example, the set of its rational points)? Is its completion the projective line?
References, if easy, would be also welcome (I have no background in algebraic geometry).
By rewriting the condition, we see that $(x',y') \sim (x,y)$ if and only if $$ x' = \frac{\alpha}{\beta} x \qquad \text{ and } \qquad y' = \frac{\alpha}{\beta} y $$ for some $\alpha,\beta \in \mathbb Z$. But if $\alpha,\beta$ ranges through all nonzero integers, then the fraction ranges through all nonzero fractions in $\mathbb Q^\times$.
Hence the space $(\mathbb Z^2 \backslash (0,0))/ \sim $ correspond exactly to points on $\mathbb P^1_{\mathbb Q}$, the projective line over the rationals.
(To see this: one direction is just by the explanation above. For the other direction, suppose given a point $[p:q] \in \mathbb P^1$. By multiplying by denominators, we can assume $p,q$ are integers. This gives a point in $\mathbb Z^2$, and this is unique up to the equivalence relation.)