2 00 9 Combinatorial and Model - Theoretical Principles Related to Regularity of Ultrafilters and Compactness of Topological Spaces

We discuss the existence of complete accumulation points of sequences in products of topological spaces. Then we collect and generalize many of the results proved in Parts I, II and IV. The present Part VI is complementary to Part V to the effect that here we deal, say, with uniformity, complete accumulation points and κ-(λ)-compactness, rather than with regularity, [λ, μ]compactness and κ-(λ, μ)-compactness. Of course, if we restrict ourselves to regular cardinals, Parts V (for λ = μ) and Part VI essentially coincide. See Parts I IV [L7], or [BF, C2, CK, CN, KM, KV, L1, L2, L3, L4, L5, L6, S, V1] for unexplained notation. Let us recall the definition of the 2 product. If ν is a cardinal, and (Xβ)β∈κ is a family of topological spaces, then the 2 <ν topology on the cartesian product ∏ β∈κXβ is the topology a base of which is given by all products ∏ β∈κ Yβ, where each Yβ is an open subset of Xβ, and |{β ∈ κ|Yβ 6= Xβ}| < ν. The product of (Xβ)β∈κ with the 2 <ν topology shall be denoted by 2 β∈κXβ. Notice that in the case ν = ω we get the more usual Tychonoff product. As usual, we shall denote the Tychonoff product by ∏ β∈κXβ. Recall that, for every infinite cardinal λ, a topological space X is said to satisfy CAPλ if and only if every subset Y ⊆ X with |Y | = λ has a complete accumulation point in X. 2000 Mathematics Subject Classification. Primary 03C20, 54D20, 03E05, 03C95, 03E75; Secondary 54B10, 54A35, 54F05, 03C55, 03C98.