I am looking for natural families of Hausdorff topologies (metrics, norms, if possible) for the space of rational functions of a single complex variable of arbitrary, unbounded denominator degree (and, if needed, a finite limit for $z\to\infty$). Or spaces of meromorphic functions containing these functions.

The main requirement is that any sequence of functions $r_n$ defined by $r_n(z):=a_n/(b_n-z)$ for $z$-independent $a_n\to a$ and $b_n\to b$ converges to to the function $r$ defined by $r(z):=a/(b-z)$.

If possible, I'd like to have a metric which gives a nice formula for the distance of $a/(b-z)$ to $a'/(b'-z)$.