Graph spaces and $\bot$-free Boolean algebras

Abstract:

Let X denote an arbitrary second-countable, compact, zero-dimensional space. Our main result says that X is a graph space, i.e., homeomorphic to the space of all complete subgraphs of a suitable graph. We first characterize graph spaces in terms of the Boolean algebras of their clopen subsets. Then it is proved that each countable Boolean algebra has the corresponding property. As a corollary we obtain that X is homeomorphic to the underlying space of a subalgebra of ${\langle {\mathbf {2}};F\rangle ^\omega }$, where 2 is the discrete two-point space and F any set of finitary operations on 2 such that neither the negation nor the ternary sum $x \oplus y \oplus z$ (addition modulo 2) belongs to the clone generated by F.