Adventures in Terminology

I think the worst terminology in all of mathematics may be that of G-delta and F-sigma sets. Even though I learned the meaning of them years ago, they still convey no information to me at first glance. I just looked at the definitions of perfectly normal space and Baire set, and without concentrating on that mysterious G-delta they mean nothing to me. Good terminology should provide a hint as why the definition isn’t some other way. Why not F-sigma sets? G-delta-sigma sets? G-sigma-delta sets?

Which is too bad, because in both instances the property is completely natural. The perfectly normal spaces are exactly those spaces where every closed set is a zero set of a continuous function. The Baire sets are for doing measure theory when you’re outside the friendly confines of a perfectly normal space. To integrate real-valued bounded continuous functions, we only need to consider Baire sets. Other sets are uninteresting from the point of view of measure theory. But in both cases it’s easier to work back from the desired property to the definition than vice versa.