Sub-cellular, neuronal and mental manifestations of systemic structure of memory

Abstract

We study the separable complementation property (SCP) and its natural variations in Banach spaces of continuous functions over compacta KA induced by almost disjoint families A of countable subsets of uncountable sets. For these spaces, we prove among other things that C(KA) has the controlled variant of the separable complementation property if and only if C(KA) is Lindelöf in the weak topology if and only if KA is monolithic. We give an example of A for which C(KA) has the SCP while KA is not monolithic and an example of a space C(KA) with controlled and continuous SCP which has neither a projectional skeleton nor a projectional resolution of the identity. Finally, we describe the structure of almost disjoint families of cardinality ω1 which induce monolithic spaces of the form KA: they can be obtained from countably many ladder systems and pairwise disjoint families by applying simple operations.