Chains and discrete sets in zero-dimensional compact spaces

Abstract

Let X X be a compact zero-dimensional space and let B ( X ) B(X) denote the Boolean algebra of all clopen subsets of X X . Let κ \kappa be an infinite cardinal. It is shown that if B ( X ) B(X) contains a chain of cardinality κ \kappa then X × X X \times X contains a discrete subset of cardinality κ \kappa . This complements a recent result of J. Baumgartner and P. Komjath relating antichains in B ( X ) B(X) to the π \pi -weight of X X .