Paul Bankston

Reduced coproducts of compact Hausdorff spaces

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.

Taxonomies of model-theoretically defined topological properties

A topological classification scheme consists of two ingredients: (1) an abstract class of topological spaces; and (2) a “taxonomy”, i.e. a list of first order sentences, together with a way of assigning an abstract class of spaces to each sentence of the list so that logically equivalent sentences are assigned the same class. is then endowed with an equivalence relation, two spaces belonging to the same equivalence class if and only if they lie in the same classes prescribed by the taxonomy. A space X in is characterized within the classification scheme if whenever Y ∊ and Y is equivalent to X , then Y is homeomorphic to X . As prime example, the closed set taxonomy assigns to each sentence in the first order language of bounded lattices the class of topological spaces whose lattices of closed sets satisfy that sentence. It turns out that every compact two-complex is characterized via this taxonomy in the class of metrizable spaces, but that no infinite discrete space is so characterized. We investigate various natural classification schemes, compare them, and look into the question of which spaces can and cannot be characterized within them.

A Hierarchy of Maps Between Compacta

Let CH be the class of compacta (i.e., compact Hausdorff spaces), with BS the subclass of Boolean spaces. For each ordinal az and pair (Km L) of subclasses of CH, we define Lev>,, (K. L). the class of maps of level at least az from spaces in K to spaces in L. in such a way that. for finite a.. Lev>,, (BS. BS) consists of the Stone duals of Boolean lattice embeddings that preserve all prenex first-order formulas of quantifier rank az. Maps of level > 0 are just the continuous surjections. and the maps of level > 1 are the co-existential maps introduced in [8]. Co-elementary maps are of level > az for all ordinals az: of course in the Boolean context, the co-elementary maps coincide with the maps of level > co. The results of this

Model-theoretic characterizations of arcs and simple closed curves

Two compact Hausdorff spaces are co-elementarily equivalent if they have homeomorphic ultracopowers; equivalently if their Banach spaces of continuous real-valued functions have isometrically isomorphic Banach ultrapowers (or, approximately satisfy the same positive-bounded sentences). We prove here that any locally connected compact metrizable space co-elementarily equivalent with an arc (resp. a simple closed curve) is itself an arc (resp. a simple closed curve). The hypotheses of metrizability and local connectedness

Some Applications of the Ultrapower Theorem to the Theory of Compacta

The ultrapower theorem of Keisler and Shelah allows such model-theoretic notions as elementary equivalence, elementary embedding and existential embedding to be couched in the language of categories (limits, morphism diagrams). This in turn allows analogs of these (and related) notions to be transported into unusual settings, chiefly those of Banach spaces and of compacta. Our interest here is the enrichment of the theory of compacta, especially the theory of continua, brought about by the importation of model-theoretic ideas and techniques.

Co-elementary equivalence, co-elementary maps, and generalized arcs

By a generalized arc we mean a continuum with exactly two non-separating points; an arc is a metrizable generalized arc. It is well known that any two arcs are homeomorphic (to the real closed unit interval); we show that any two generalized arcs are co-elementarily equivalent, and that co-elementary images of generalized arcs are generalized arcs. We also show that if f : X → Y is a function between compacta and if X is an arc, then f is a co-elementary map if and only if Y is an arc and f is a monotone continuous surjection.

Notions of Relative Ubiquity for Invariant Sets of Relational Structures

Given a finite lexicon L of relational symbols and equality, one may view the collection of all L-structures on the set of natural numbers ω as a space in several different ways. We consider it as: (i) the space of outcomes of certain infinite two-person games; (ii) a compact metric space; and (iii) a probability measure space. For each of these viewpoints, we can give a notion of relative ubiquity, or largeness, for invariant sets of structures on ω

The Chang-Łoś-Suszko theorem in a topological setting

Abstract.The Chang-Łoś-Suszko theorem of first-order model theory characterizes universal-existential classes of models as just those elementary classes that are closed under unions of chains. This theorem can then be used to equate two model-theoretic closure conditions for elementary classes; namely unions of chains and existential substructures. In the present paper we prove a topological analogue and indicate some applications.

Co-elementary equivalence for compact Hausdorff spaces and compact abelian groups

Abstract The notions of elementary equivalence and elementary mapping in first order model theory have category-theoretic reflections in many well-known topological settings. We study the dualized notions in the categories of compact Hausdorff spaces and compact abelian groups.

Minimal freeness and commutativity

Apseudobasis for an abstract algebraA is a subsetX ofA such that every mappingX intoA extends uniquely to an endomorphism onA. A isminimally free ifA has a pseudobasis. In this paper we look at how minimal freeness interacts with various notions of commutativity (e.g., “operational” commutativity in the algebra, usual commutativity in the endomorphism monoid of the algebra). One application is a complete classification of minimally free torsion abelian groups.