Topological Reduced Products and the GCH
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Since its inception in the late fifties, the theory of reduced products in model theory and algebra has developed into an active field of research with increasingly many participants. In particular the theory of ultraproducts has provided "algebraic" proofs of the compactness theorem of first order logic, the existence of saturated models of certain kinds; as well as a characterization of the notion of elementary equivalence between models. Copious details can be found in [BS] and [CK]. In our paper [B] we attempted to translate the notion of reduced product into the context of general topology and found, not too surprisingly, that here was a vast untapped source of research problems, many of the type already encountered in the theory of box products. In [B] several parallel problems involving box products and "topological" ultraproducts are explored; and it turns out that the ultraproduct theorems are often either easier than their counterparts to prove or can be proved directly in ZFC without recourse to extra set-theoretic axioms. Topological reduced products are formed as certain quotients of box products where the equivalence relations in question derive from filters on the index set. In this note ~N'e present a result about paracompactness in topological ultraproducts (i.e. where the filter is maximal) and show how this result relates to a known theorem about paracompactness in box products (trivially reduced products via the singleton filter). Both of these results relate directly with the Generalized Continuum Hypothesis (GCH).
Bankston, Paul, "Topological Reduced Products and the GCH" (1976). Mathematics, Statistics and Computer Science Faculty Research and Publications. 187.