A hierarchy of maps between compacta

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 $(K,L)$ of subclasses of $CH$, we define $Le{v}_{>=alpha}(K,L)$, the class of maps of level at least $alpha$ from spaces in $K$ to spaces in $L$, in such a way that, when $alpha<omega$, Lev}_{>=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 $>=0$ are just the continuous surjections, and the maps of level $>=1$ are the co-existential maps. Co-elementary maps are of level $>=alpha$ for all ordinals $alpha$; of course in the Boolean context, the co-elementary maps coincide with the maps of level $>=omega$. The results of this paper include: (i) every map of level $>=omega$ is co-elementary; (ii) the limit maps of an $omega$-indexed inverse system of maps of level $>=alpha$ are also of level $>=alpha$; and (iii) if $K$ is a co-elementary class, $k<omega$ and $Lev_{>=k}(K,K) = Lev_{>=k+1}(K,K)$,then$Lev_{>=k}(K,K}) = Lev_{>=omega}(K,K)$. A space $X$ in $K$ is co-existentially closed in $K$ if $Lev_{>=0}(K,{X}) = Lev_{>=1}(K,{X})$. We showed in an earlier paper 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.