A Hierarchy of Maps Between Compacta

This paper, a continuation of [2]–[8], aims to carry on the project of establishing model-theoretic concepts and methods within the topological context; namely that of compacta (i.e., compact Hausdorff spaces). Since there is a precise duality between the categories of compacta (plus continuous maps) and commutative B-algebras (plus nonexpansive linear maps) (the Gel’fand-Năımark theorem [21]), our enterprise may also be seen as part of Banach model theory (see [12]–[16]). The main difference is that we are doing Banach model theory “in the mirror,” so to speak, and it is often the case that a mirror can help one focus on features that might otherwise go unnoticed. In the interests of being as self-contained as possible, we present a quick review of our main tool, the topological ultracoproduct construction. It is this construction, plus the landmark ultrapower theorem of Keisler-Shelah [9], that gets our project off the ground. (Detailed accounts may be found in [2]–[6] and [11].) We let CH denote the category of compacta and continuous maps. In model theory, it is well known that ultraproducts may be described in the language of category theory; i.e., as direct limits of (cartesian) products, where the directed set is the ultrafilter with reverse inclusion, and the system of products consists of cartesian products taken over the various sets in the ultrafilter. (Bonding maps are just the obvious restriction maps.) When we transport this framework to the category-opposite of CH, the result is the topological ultracoproduct (i.e., take an inverse limit of coproducts), and may be concretely described as follows: Given compacta 〈Xi : i ∈ I〉 and an ultrafilter D on I, let Y be the disjoint union ⋃ i∈I(Xi×{i}) (a locally compact space). With q : Y → I the natural projection onto the second coördinate (where I has the discrete topology), we then have the Stone-Čech lifting q : β(Y ) → β(I). Now the ultrafilter D may be naturally viewed as an element of β(I), and it is not hard to show that the topological ultracoproduct ∑ DXi is the pre-image (q )[D]. (The reader