@OwenBiesel Yes, because $z_1, z_2$ are (positive or negative) integers.

Thanks! - Is it also possible to find an equivalence relation $R$ on a set $X$ such that there is no topology $\tau$ on $X$ such that $\mathrm{cl}(\Delta_X) = R$?

That's correct - I put the absolute value signs in the wrong place and edited the post now accordingly.

Sorry - I really got my $|\cdot|$ signs wrong. What I intended is the inequality $|\frac{m}{n} - (u'+v')| < |\frac{m}{n} - (u+v)|$. The goal is to approximate $\frac{m}{n}$ as good as it gets by a sum or difference of unit fractions, so I want to minimize $|\frac{m}{n} - (u+v)|$ where $u,v\in U$.

That's right - I just edited my post accordingly. Thanks!

About Qiaochu's question: >> Is it obvious that there exists a compact Hausdorff topology on every set? << Let $X\neq \emptyset$ be a set, fix $x_0\in X$. Let $\tau = \mathcal{P}(X\setminus\{x_0\}) \cup \{U\subseteq X : X\setminus U \textrm{ is finite }\}$. Then $\tau$ is a compact Hausdorff topology on $X$.