I’m looking for some information on the topology of “the”* hyperreals (actually, what I believe I really need is topology of just the finite hyperreals specifically, but either way should be good).
*(I’m putting “the” in quotation marks since I figure the structure of the hyperreals depends on the choice of the ultrafilter used in the construction, giving many different sets of hyperreals).
In particular, I’d like to know:
A set of properties that completely characterizes the topology of the (finite) hyperreals.
Can different ultrafilters result in nonhomeomorphic hyperreal topologies? (Does the topology of the hyperreals depend on the ultrafilter used in the construction?)
(The following is from a CS guy, not a Math guy, so take it just as questions being asked, not as complaints.)
What do you mean by “finite”? If you mean Reals+infinitesimals then don’t you have closure problems? If you exclude all the infinitesimals (except 0) then don’t you have the Reals?
ftg, by “finite” I mean the reals plus infinitesimals. There’s no closure problems because I’m only interested in the topological structure, not the algebraic structure (just like you can talk of the topological structure of any subset of the reals, whether or not that subset has any sort of algebraic closure. Any subset of a topological space automatically inherits a topology).
By the way, I’m not necessarily expecting any one to come and directly answer my questions. If anyone could point me to some text or paper which focuses on the topological structure, that would be enough.
Just a bump to the top, along with one thing I learned today, for anyone else that might be interested.
Today I found out that under the continuum hypothesis, the choice of ultrafilter is irrelevant. Under CH, a construction of the hyperreals using one nonprincipal ultrafilter is isomorphic (as an ordered field) to the hyperreals constructed using any other nonprincipal ultrafilter.
The text I got this from (Lectures on the Hyperreals: An Introduction to Nonstandard Analysis by Robert Goldblatt) also says, “Without this assumption (CH) the situation is undetermined”.
Any metrizable space is first countable (every point in the space has a countable neighborhood base). The hyperreals, however, are not first countable.
And by the way, I want to thank you for asking that question, ultrafilter. I wasn’t sure whether the hyperreals were first countable or not, so that prompted me to do a quick search to find out. I found this page, which actually answers the question that prompted this thread in the first place. (I was wondering whether it was possible a particular continuum I was looking at was homeomorphic to “the” hypereals. It can’t be, since the hyperreals are totally disconnected).
I haven’t really looked at this and am not an analyst by trade, but my first guess would be that this looks something like the “long line”. In that case it’s the failure of local compactness that makes the space nonmetrizable.
Dammit, you’re right. It’s been a long time since I’ve pulled out Hocking and Young to do any point-set topology. My best guess as a patch would be a countability argument, similar to the hyperreals. That other link posted earlier shows that the hyperreals are definitely not the long line.
Honest, give me a nice, sensible manifold or a homology theory and I’m there. This point-set stuff, though…
Which is why, despite my username, I haven’t had anything to add to this thread.
This Wikipedia page points out that manifolds are metrizable iff paracompact, and the long line isn’t. Actually, I was taught that the sensible thing to do was include paracompactness as part of the definition of a manifold, to rule out spaces like the long line.
–Topologist (algebraic by preference, point-set under duress)
Actually, even though the hyperreals are not locally compact, that doesn’t demonstrate that they are nonmetrizable. For example, the rationals are not locally compact, yet they are metrizable.
Thanks for the suggestion, Tyrrell McAllister. I am still wondering what the Dedekind completion of the finite hyperreals looks like (i.e., use Dedekind cuts to extend the set of ordered hyperreals so that it has the least upper bound property).
