2 00 8 Combinatorial and Model - Theoretical Principles Related to Regularity of Ultrafilters and Compactness of Topological Spaces

We find many conditions equivalent to the modeltheoretical property λ κ ⇒ μ introduced in [L1]. Our conditions involve uniformity of ultrafilters, compactness properties of products of topological spaces and the existence of certain infinite matrices. See Part I [L7] or [CN, CK, KM, KV, HNV] for unexplained notation. According to [L1], if λ ≥ μ are infinite regular cardinals, and κ is a cardinal, λ κ ⇒ μ means that the model 〈λ,<, γ〉γ<λ has an expansion A in a language with at most κ new symbols such that whenever B ≡ A and B has an element x such that B |= γ < x for every γ < λ, then B has an element y such that B |= α < y < μ for every α < μ. An ultrafilter D over λ is said to be uniform if and only if every member of D has cardinality λ. If λ is a regular cardinal, then it is obvious that an ultrafilter D is uniform over λ if and only if the interval [0, γ] 6∈ D, for every γ < λ, if and only if the interval (γ, λ) is in D, for every γ < λ. Thus, if D is an ultrafilter over some regular cardinal λ, and if IdD denotes the D-class of the identity function on λ, then D is uniform over λ if and only if in the model C = ∏ D A we have that d(γ) < IdD for every γ < λ. Here, d denotes the elementary embedding. IfD is an ultrafilter over I, and f : I → J , then f(D) is the ultrafilter over J defined by: Y ∈ f(D) if and only if f(Y ) ∈ D. If κ, λ are infinite cardinals, a topological space is said to be [κ, λ]compact if and only if every open cover by at most λ sets has a subcover 2000 Mathematics Subject Classification. Primary 03C20, 03E05, 54B10, 54D20; Secondary 03C55, 03C98.