Reduced coproducts of compact Hausdorff spaces

Abstract

By analyzing how one obtains the Stone space of the reduced product of an indexed collection of Boolean algebras from the Stone spaces of those algebras, we derive a topological construction, the “reduced coproduct”, which makes sense for indexed collections of arbitrary Tichonov spaces. When the filter in question is an ultrafilter, we show how the “ultracoproduct” can be obtained from the usual topological ultraproduct via a compactification process in the style of Wallman and Frink. We prove theorems dealing with the topological structure of reduced coproducts (especially ultracoproducts) and show in addition how one may use this construction to gain information about the category of compact Hausdorff spaces.

The Chang-Łoś-Suszko theorem in a topological setting

“…Also, because the ultracoproduct operation on subsets commutes with finite unions and intersections, and because (see Proposition 1.5 in [1]) the Boolean lattice of clopen subsets of an ultracoproduct of compacta is the corresponding ultraproduct of the clopen set lattices of those compacta, we infer that…”

“…Given x i : i ∈ I ∈ i∈I X i , there is just one point of D X i containing D {x i } as an element; call this point D x i . Then, by basic results in [1], the set of such points is dense in D X i . In light of this, we fix D x i ∈ H \ K and D y i ∈ K \ H.…”

“…Proof. By Theorem 2.6 in [5], the class of compacta of covering dimension ≤ n is closed under co-existential maps; by Theorem 2.2.2 in [1], it is closed under ultracopowers. A nearly identical argument shows the class to be closed under all ultracoproducts.…”

“…What makes Theorems 2.6 in [5] and 2.2.2 in [1] work is the theorem of E. Hemmingsen (Lemma 2.2, and its corollary, in [9]) to the effect that a normal Hausdorff space X has covering dimension ≤ n if and only if, whenever {B 1 , . .…”

“…The most flexible way to form D X i for our present purposes is to take the following steps (see [1,3]):…”