$\hbox{Rat}_d$, the space of rational functions of degree $d$, is a $2d+1$ dimensional affine algebraic variety, so it embeds into $\mathbb A^N$ for some suffficiently large $N$. Since you're presumably interested in working over $\mathbb C$ (or maybe $\mathbb R)$, you can embed $\hbox{Rat}_d(\mathbb C)$ into $\mathbb C^N$ and then just use the usual topology on $\mathbb C^N$.
To be more precise, if you view a degree $d$ rational function $f(z)=F(z)/G(z)$ as the $2d+2$-tuple of the coefficients of $F$ and $G$, it is defined as a point in $\mathbb P^{2d+2}$, and $\hbox{Rat}_d$ is the complement of the resultant locus $\mathcal R:\hbox{Res}(F,G)=0$. The complement of a hypersurface such as $\mathcal R$ is an affine variety. There's a brief description of how this embedding works in (see especially Proposition 4.27)
The Arithmetic of Dynamical Systems, Springer, Section 4.3 "The Space $\hbox{Rat}_d$ of Rational Functions"