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Association for Symbolic Logic
Journal of Symbolic Logic
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Shelves: BC1 .J6 Raynor Memorial Periodicals
Let CH be the class of compacta (i.e., compact Hausdorff spaces), with BS the subclass of Boolean spaces. For each ordinal α and pair $\langle K,L\rangle$ of subclasses of CH, we define Lev≥α K,L), the class of maps of level at least α from spaces in K to spaces in L, in such a way that, for finite α, Lev≥α (BS,BS) consists of the Stone duals of Boolean lattice embeddings that preserve all prenex first-order formulas of quantifier rank α. Maps of level ≥ 0 are just the continuous surjections, and the maps of level ≥ 1 are the co-existential maps introduced in . Co-elementary maps are of level ≥α for all ordinals α; of course in the Boolean context, the co-elementary maps coincide with the maps of level ≥ω. The results of this paper include: (i) every map of level ≥ω is co-elementary; (ii) the limit maps of an ω-indexed inverse system of maps of level ≥α are also of level ≥α; and (iii) if K is a co-elementary class, k < ω and Lev≥ k(K,K) = Lev≥ k+1 (K,K), then Lev≥ k(K,K) = Lev≥ω(K,K). A space X ∈ K is co-existentially closed in K if Lev≥ 0(K, X) = Lev≥ 1 (K, X). Adapting the technique of "adding roots," by which one builds algebraically closed extensions of fields (and, more generally, existentially closed extensions of models of universal-existential theories), we showed in  that every infinite member of a co-inductive co-elementary class (such as CH itself, BS, or the class CON of continua) is a continuous image of a space of the same weight that is co-existentially closed in that class. We show here that every compactum that is co-existentially closed in CON (a co-existentially closed continuum) is both indecomposable and of covering dimension one
Bankston, Paul, "A Hierarchy of Maps Between Compacta" (1999). Mathematics, Statistics and Computer Science Faculty Research and Publications. 138.
Published version. The Journal of Symbolic Logic, Vol. 64, No. 4 (December 1999): 1628-1644. Publisher Link. © 1999 The Association for Symbolic Logic. Used with permission.