Mittag-Leffler condition: what's the origin of its name?

Why the Mittag-Leffler condition on a short exact sequence of, say, abelian groups, that ensures that the first derived functor of the inverse limit vanishes, is so named?

It turns out - or so I'm told, I must admit to never working through the details - that parts of the proof can be abstracted, and from this point of view a key ingredient (implicit or explicit in the proof, according to taste) is the vanishing of a certain $\lim_1$ group -- as assured by the "abstract" ML-theorem that you mention.
• Thanks Yemon. I followed your suggestion and took a look at Runde's (Appendix A) and the "abstract" Bourbaki's M-L version for a sequence of complete metric spaces and continuous funcions $f_n:X_n\rightarrow X_{n-1}$ with dense image. It looks that this can be further abstracted to the "algebraic" M-L. I'll do the details later on to see if this is the case. By the way, Runde quotes Bourbaki's "General Topology" volume. I don't have access to that one right now but I'll check it tomorrow. Thanks again. – F Zaldivar Feb 9 '10 at 1:26