525 research outputs found

    A Hierarchy of Maps Between Compacta

    Get PDF
    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 ⟨K,L⟩\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 [8]. 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 \u3c 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 [8] 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 on

    Combinatorial Properties and Dependent choice in symmetric extensions based on L\'{e}vy Collapse

    Get PDF
    We work with symmetric extensions based on L\'{e}vy Collapse and extend a few results of Arthur Apter. We prove a conjecture of Ioanna Dimitriou from her P.h.d. thesis. We also observe that if VV is a model of ZFC, then DC<κDC_{<\kappa} can be preserved in the symmetric extension of VV in terms of symmetric system ⟨P,G,F⟩\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle, if P\mathbb{P} is κ\kappa-distributive and F\mathcal{F} is κ\kappa-complete. Further we observe that if VV is a model of ZF + DCκDC_{\kappa}, then DC<κDC_{<\kappa} can be preserved in the symmetric extension of VV in terms of symmetric system ⟨P,G,F⟩\langle \mathbb{P},\mathcal{G},\mathcal{F}\rangle, if P\mathbb{P} is κ\kappa-strategically closed and F\mathcal{F} is κ\kappa-complete.Comment: Revised versio

    Laver and set theory

    Full text link
    In this commemorative article, the work of Richard Laver is surveyed in its full range and extent.Accepted manuscrip
    • …
    corecore