There should really be a book -- it could be called, say, "Topological and Ordered Algebraic Structures" -- which covers, like, all the basic properties of topological STUFF and ordered STUFF (groups, abelian groups, rings, fields, modules, vector spaces...) and, in particular, what happens when you complete them (and what structure the completion has).
Seriously, you can get out a basic algebra book like Dummit & Foote or something and learn pretty much all the basics of how just the simple algebraic versions work, but where is there a unified source for the topological versions? Even if it just contained statements and references without proofs, that would be helpful. And it could, like, presume prior knowledge of uniform spaces, because hey you should probably know that first if you want to think about these things. And state when things are unknown[4].
So far I've only seen this stuff in scattered sources, which are often quite incomplete, and often aren't as general as they could be (I get the impression lots of people care about topological vector spaces, but I think mostly over R and C). Really I'm asking for something that doesn't focus on the serious math most people care about, but rather about the obvious very general abstract questions that just kind of bug you. And it could go into more serious stuff where results exist, but mostly I just want something that thoroughly covers all the basic questions you're going to think to ask first, especially about the nicest cases (like say when the topology comes from an ordering, or a "generalized absolute value", or something, and there's a lot of algebraic structure...).
Maybe such a book exists? I would like to know of it if it does, because these things are bugging me. No, this has nothing to do with any of my actual work (which maybe I might get back to actually writing about in, uh, a few months :P )...
This reminds me -- I meant to ask on MathOverflow, what's the most general setting for "continuity of roots"? The roots of a polynomial vary continuously in the coefficients, so long as the leading coefficient doesn't become 0, right? Of course the problem is finding a correct formal statement of that, and also, well, proving it -- the various proofs I've seen have all been for particular special cases; never any nice general statement. What's up with that? (Similarly with statements that "topological completions of things which are complete in some algebraic sense are again complete in that algebraic sense", which I think is typically proved via some sort of continuity of roots...)
Yeah, I should actually go ask that.
-Harry
[0]This isn't really an example of something being *hard*, but I still think it's a worthwhile example -- if you have a balanced group[3], proving that its completion is again a topological group is a bit of easy abstract nonsense. But if you have a topological *ring*, proving that its completion naturally has the structure of a topological ring actually requires real proof.
[3]For general topological groups, the completion is not again a group! Of course, for general topological groups, you have to ask, "the completion with respect to which uniformity"...
[4]Here's an example: The completion of a topological field is not necessarily again a field (though if it's a field, it's a topological field). As of, uh, 1988, all known examples of this have zero divisors; is it possible to get an integral domain that isn't a field? Well, like I said, as of 1988 this is unknown... I kind of dread trying to find a more up-to-date source on this... I most likely won't try because, do I care that much? :-/
