On the Property “sep” of Partial Orderings

A partial ordering P has the property SEP (or SEP(P )), if, for a sufficiently large regular cardinal χ, the family of elementary submodels of H(χ) of cardinality א1 with the property that [M ]א0 ∩M is cofinal subset (with respect to ⊆) of [M ]א0 and P ∩M ≤σ P , is cofinal (also with respect to ⊆) in [H(χ)]א1 (see Proposition 2 below). We describe, in section 1, against which historical background this notion came to be formulated. In section 2, we introduce the property SEP and some other related properties of partial orderings P such as the weak Freese-Nation property and (א1,א0)-ideal property and review some known results around these properties. In section 3 we give a sketch of a proof of the theorem asserting that the combinatorial principle Princ introduced by S. Shelah does not imply SEP of 〈P(ω),⊆〉. The following is based on author’s talk at General Topology Symposium 2002 held on 18–20, November 2002 at Kobe University, Japan. The property SEP to be introduced in section 2 will be the main subject of this article. But since this is going to appear in the proceedings of the general topology symposium, let us begin with a historical account explaining the connection to topology. 1. Historical background Remember that a zero-dimensional Hausdorff space is also called a Boolean space — a topological space X is said to be zero-dimensional if closed and open (clopen) subsets of X constitute a topological base of X. It is well-known that there are contravariant functors between the category of Boolean algebras and the category of Boolean spaces (Stone Duality Theorem). By the duality, a Boolean algebra B is related to the space Ult(B) of all ultra-filters on B with a canonical topology and a Boolean space X is related to the Boolean algebra Clop(B) of clopen subsets of X partially ordered by ⊆ (see e.g. [16]). One of the most important classes of Boolean spaces is that of the generalized Cantor spaces 2 which are the product spaces of κ copies of the discrete space 2 = {0, 1} with their Boolean algebraic dual being free Boolean algebras Fr(κ) with a free generator of size κ. A variety of classes of topological spaces which are more or less similar to the generalized Cantor spaces, such as dyadic spaces, Dugundji spaces and κ-metrizable spaces, have been studied extensively in the literature (see e.g. the reference of [17] and [12]). These classes of topological spaces have natural counterparts in the category of Boolean algebras via Stone Duality Theorem (see [17], [12]):