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, or zero, 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 $a\_n\to a\ne 0$ 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 $a'/(b'-z)$.



Natural topologies for the space of rational functions
Karl-Goswin Grosse-Erdmann, The locally convex topology on the space of meromorphic functions, J. Austr. Math. Soc., 59 (1995) 287-303


