## Journal of Symbolic Logic

### A Hierarchy of Maps between Compacta

Paul Bankston

#### Abstract

Let CH be the class of compacta (i.e., compact Hausdorff spaces), with BS the subclass of Boolean spaces. For each ordinal $\alpha$ and pair $\langle K,L\rangle$ of subclasses of CH, we define $Lev_{\geq\alpha} K,L)$, the class of maps of level at least $\alpha$ from spaces in K to spaces in L, in such a way that, for finite $\alpha$, $Lev_{\geq\alpha}$ (BS,BS) consists of the Stone duals of Boolean lattice embeddings that preserve all prenex first-order formulas of quantifier rank $\alpha$. Maps of level $\geq$ 0 are just the continuous surjections, and the maps of level $\geq$ 1 are the co-existential maps introduced in [8]. Co-elementary maps are of level $\geq\alpha$ for all ordinals $\alpha$; of course in the Boolean context, the co-elementary maps coincide with the maps of level $\geq\omega$. The results of this paper include: (i) every map of level $\geq\omega$ is co-elementary; (ii) the limit maps of an $\omega$-indexed inverse system of maps of level $\geq\alpha$ are also of level $\geq\alpha$; and (iii) if K is a co-elementary class, k < $\omega$ and $Lev_{\geq k}(K,K)$ = $Lev_{\geq k+1} (K,K)$, then $Lev_{\geq k}(K,K)$ = $Lev_{\geq\omega}(K,K)$. A space X $\in$ K is co-existentially closed in K if $Lev_{\geq 0}(K, X)$ = $Lev_{\geq 1} (K, X)$. 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.

#### Article information

Source
J. Symbolic Logic, Volume 64, Issue 4 (1999), 1628-1644.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183745943

Mathematical Reviews number (MathSciNet)
MR1780075

Zentralblatt MATH identifier
0962.03024

JSTOR