On character and chain conditions in images of products

A scadic space is a Hausdorff continuous image of a product of compact scattered spaces. We complete a theorem begun by G. Chertanov that will establish that for each scadic space X, χ(X) = w(X). A ξ-adic space is a Hausdorff continuous image of a product of compact ordinal spaces. We introduce an either-or chain condition called Property ${\displaystyle R_{\lambda }\prime }$ which we show is satisfied by all ξ-adic spaces. Whereas Property ${\displaystyle R_{\lambda }\prime }$ is productive, we show that a weaker (but more natural) Property ${\displaystyle R_{\lambda }}$ is not productive. Polyadic spaces are shown to satisfy a stronger chain condition called Property ${\displaystyle R_{\lambda }\prime \prime }$. We use Property ${\displaystyle R_{\lambda }\prime }$ to show that not all compact, monolithic, scattered spaces are ξ-adic, thus answering a question of Chertanov's.