Abstract
In the sense used here a frame A is the algebraic generalization of a topology, the family of open sets of a topological space. It is a complete Heyting algebra, although from that perspective frame morphisms are not quite what you expect. The category of complete Boolean algebras sits inside the category of frames. However, a frame need not have a Boolean reflection. It seems that it does have a Boolean reflection precisely when it is ‘nearly pathological’ in some sense. For instance, the topology of a \(T_0\) space has Boolean reflection precisely when the space is scattered. Each frame A has an assembly NA which collects together all the quotients of A, and this assembly is itself a frame. Since NA is a frame it has its own assembly \({{N}^2}A\), which has its assembly \({{N}^3}A\), and so on. This generates the assembly tower of A. It is known that A has Boolean reflection precisely when some member of this tower is Boolean, and then that is the Boolean reflection. It seems that the nature of this tower is somehow connected with a generalization of the Cantor-Bendixson process on a topological space. In this chapter I investigate this idea.
In memory of Leo Esakia
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Simmons, H. (2014). Cantor-Bendixson Properties of the Assembly of a Frame. In: Bezhanishvili, G. (eds) Leo Esakia on Duality in Modal and Intuitionistic Logics. Outstanding Contributions to Logic, vol 4. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-8860-1_9
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