It’s a sad truth, but the mathematics research literature is very tough going for beginners. By “beginners” I mean bright high-school students, or university students, or beginning graduate students, or even professional mathematicians who are trained in an area different from the article he/she is trying to read. […]
Things like this permeate the mathematical literature. Take compactness, for example. Compactness is a marvelous tool that lets you deduce — usually in a non-constructive fashion — the existence of objects (particularly infinite ones) from the existence of finite “approximations”. Formally, compactness is the property that a collection of closed sets has a nonempty intersection if every finite subcollection has a nonempty intersection; alternatively, if every open cover has a finite subcover.
Now compactness is a topological property, so to use it, you really should say explicitly what the topological space is, and what the open and closed sets are. But mathematicians rarely, if ever, do that. In fact, they usually don’t specify anything at all about the setting; they just say “by the usual compactness argument” and move on. That’s great for experts, but not so great for beginners.
I really wonder who was the very first to take this particular lazy approach to mathematical exposition.
Hmmm, often it is the reviewer, referee or editor. They accept your paper, but make you take out some details (and, to be fair, add others)
A colleague and I are thinking of starting a journal called “The Journal of Omitted Details”.
But yes, this practice makes some mathematics very difficult for the non-expert to read.
Note: the usual definition of a compact set (given some topology) is: is compact if, given any collection of open sets where ,there exists a finite number of the where . That is, any open cover has a finite subcover. This is equivalent to saying that any infinite set of points in has a limit point, and in a metric space this means that is both closed and bounded.