We introduce the concept of neighborhood systems as a generalization of directed, reflexive graphs and show that the prime factorization of neighborhood systems with respect to the the direct product is unique under the condition that they satisfy an appropriate notion of thinness.
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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 $R_λ'$ which we show is satisfied by all ξ-adic spaces. Whereas Property $R_λ'$ is productive, we show that a weaker (but more natural) Property $R_λ$ is not productive. Polyadic spaces are shown to satisfy a stronger chain condition called Property $R_λ''$. We use Property $R_λ'$ to show that not all compact, monolithic, scattered spaces are ξ-adic, thus answering a question of Chertanov's.
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