Chains and discrete sets in zero-dimensional compact spaces

Abstract:

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