#### Document Type

Article

#### Language

eng

#### Format of Original

17 p.

#### Publication Date

12-1999

#### Publisher

Association for Symbolic Logic

#### Source Publication

Journal of Symbolic Logic

#### Source ISSN

0022-4812

#### Original Item ID

Shelves: BC1 .J6 Raynor Memorial Periodicals

#### Abstract

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 [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 < 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 one

## Comments

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.