Shuffle products and other operations in the class of primitive spaces (Q1922116)

Primitive spaces -- independently introduced by Hanf and Pierce -- are metric Boolean spaces with a basis of clopen sets \(X\) with the property that for each clopen subset \(O\) of \(X\), either \(O \cong X\) or \(X \backslash O \cong X\). Primitive spaces can be characterized by associated diagrams (transitive relations with greatest element) that make them particularly easy to handle. In this paper, a list of usual topological constructions is given under which the class of primitive spaces is closed. Particular emphasis is laid upon the shuffle product of spaces (an operation introduced by Tarski on ordered types) and the diagram of a shuffle product is given in terms of the diagrams of the factors. The class of all primitive spaces is shown to be the smallest nonempty class of spaces closed under two special kinds of shuffle product; the obtained theorem is a topological version of a result of Heindorf.