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Ultraproducts of fields, II *June 28, 2008*

*Posted by Alexandre Borovik in Uncategorized.*

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I continue my post on ultraproducts. So, we want to understand in what sense an ultraproduct of finite fields of unbounded order is a limit at infinity of finite fields. The answer now should be obvious: since ultraproducts are residue fields for maximal ideals in the cartesian product

the topology in question should be the canonical topology (Zariski topology) of the spectrum of the ring . It instantly follows from the description of ideals and maximal ideals in that this is the Stone topology on the set of ultrafilters on , or, what is th same, the Cech-Stone compactification of the set with descrete topology. Therefore the answer is: an ultraproduct is the limit in the Cech-Stone compactification of a discrete countable set.

I have to admit that at this point I reached limits of my knowledge of set-theoretic topology and had to dip into Wikipedia. It happened that , and even more so its non-principal part is characterised by some unique properties: If the continuum hypothesis holds then is the unique Parovicenko space, up to isomorphism.

According to Wikipedia, a Parovicenko space is a topological space *X* satisfying the following conditions:

*X*is compact Hausdorff*X*has no isolated points*X*has weight*c*, the cardinality of the continuum (this is the smallest cardinality of a base for the topology).- Every two disjoint open
*F*_{σ}subsets of*X*have disjoint closures - Every nonempty
*G*_{δ}of*X*has non-empty interior.

As you can see, is uniquely characterised by very natural properties.

It is yet another manifestation of of one of the most pardoxical properties of mathematical infinity: *canonicity* of workable constructions in the infinite domain.

I wonder how much of what is natural about can be put in category theoretic terms. The you mention is a functor, Stone-Cech compactification, from Top (the category of topological spaces) to CHaus (the category of compact Hausdorff spaces). It is left adjoint to the underlying functor going the other way.

So, . In the case of a map to a compact Hausdorff space is determined by a map from to that space.

But does the formation of by removing the principal part, correspond to a category theoretic construction?