Co-elementary equivalence for compact Hausdorff spaces and compact abelian groups (Q2639052)

Two compact Hausdorff spaces X, Y are called co-elementary equivalent (we write \(X\equiv Y)\) if all ultracoproducts of X and Y are homeomorphic. It is shown that the equivalence \(\equiv\) has at most continuum classes. On the other hand there exist exactly countably many \(\equiv\)-classes consisting of 0-dimensional compact Hausdorff spaces, and for \(n>0\), there exist exactly continuum \(\equiv\)-classes consisting of n- dimensional compact Hausdorff spaces. Further every compact Hausdorff space is co-elementary equivalent to a compact metrizable space. If two completely regular spaces have co-elementary equivalent lattices of zero- sets then the Banach spaces of their bounded continuous real-valued functions approximately satisfy the same positive bounded sentences. The analogous problems for compact groups are discussed.