Journal of Symbolic Logic

A Hierarchy of Maps between Compacta

Paul Bankston

Full-text is available via JSTOR, for JSTOR subscribers. Go to this article in JSTOR.


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

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

First available in Project Euclid: 6 July 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 03C20: Ultraproducts and related constructions
Secondary: 54B35: Spectra 54C10: Special maps on topological spaces (open, closed, perfect, etc.) 54D05: Connected and locally connected spaces (general aspects) 54D30: Compactness 54D80: Special constructions of spaces (spaces of ultrafilters, etc.) 54F15: Continua and generalizations 54F45: Dimension theory [See also 55M10] 54F55: Unicoherence, multicoherence

Ultraproduct Ultracoproduct Compactum Continuum Co-Elementary Map Co-Existential Map Map of Level $\geq\alpha$


Bankston, Paul. A Hierarchy of Maps between Compacta. J. Symbolic Logic 64 (1999), no. 4, 1628--1644.

Export citation