In his book 'Handbook of Analysis and its Foundations', Schechter suggests how, for Hausdorff spaces, the Axiom of Choice may be substituted by the Ultrafilter Principle.
In particular he notices how one of the most important equivalent formulations of the Axiom of Choice, the Tychonov Product Theorem takes the following, obvious, form for the Hausdorff spaces:
An arbitrary product of compact Hausdorff spaces is compact
The surprising fact is, this is an equivalent statement to the Ultrafilter Princple, which is in general strictly weaker than the AOC.
Anyway the author does not develop further.
Now what I would like to know is:
- Can we make the substitution "UF+H=AC" more precise, in general topology?
- We do not have a notion of Hausdorfness in other context than topology, as far as I know. Can we nonetheless impose some other hypotheses on our (structured) set to substitute the AC with UF? If yes, do these hypotheses share someting with Hausdorfness?
If all you want to do is general topology / functional analysis on Hausdorff spaces (and not want to prove all ideals are contained in a maximal ideal, or want to use Zorn's lemma in another setting) you probably could get away with the UFP alone. But choosing foundations is done for all mathematicians, not just those in some subfield. It's sort of "Reverse foundations": how many and what axioms do I need to prove my result? I don't think Schechter proposes to replace AC by UFP in general, he just wants to make a point that in analysis one rarely needs full choice, and that many forms of choice one does need are equivalent to UFP. OTOH he probably does want cardinal numbers to be in a total ordering (and IIRC that's equivalent to AC?) and analysis is not an island onto itself. I wouldn't take his remarks on this too seriously.