Cardinal Invariants Related to Sequential Separability

Cardinal invariants related to sequential separability of generalized Cantor cubes 2κ, introduced by M. Matveev, are studied here. In particular, it is shown that the following assertions are relatively consistent with ZFC: (1) 2ω1 is sequentially separable, yet there is a countable dense subset of 2ω1 containing no non-trivial convergent subsequence, (2) 2ω1 is not sequentially separable, yet it is sequentially compact. The work contained in this paper is devoted to studying combinatorial properties of independent families and their relationship with sequential separability of generalized Cantor cubes. Connections with Q-sets and hence the existence of separable non-metrizable Moore spaces is also mentioned. A topological space X is sequentially separable if there is a countable D ⊆ X such that for every x ∈ X there is a sequence {xn : n ∈ ω} ⊆ D converging to x; such a D ⊆ X will be called sequentially dense in X. A space X is strongly sequentially separable if it is separable and every countable dense subset of X is sequentially dense. Here we consider sequential separability of 2 equipped with the product topology. Recall that a set A is a pseudo-intersection of a family F ⊆ [ω] if A ⊆∗ F for every F ∈ F and, F is centered if every non-empty finite subfamily of F has an infinite intersection. A family S ⊆ [ω] is splitting if ∀A ∈ [ω] ∃S ∈ S |A ∩ S| = |A \ S| = ω. A family I ⊆ [ω] is independent provided that for every nonempty disjoint F1,F2 ∈ [I]<ω F1 \ F2 6= ∅. I ⊆ [ω] is independent splitting if it is both independent and splitting. The following cardinal invariants are standard. p = min{|F| : F ⊆ [ω] a centered system which has no infinite pseudointersection} s = min{|S| : S is a splitting family} F. Tall in [Ta] showed that 2 is strongly sequentially separable for every κ < p. M. V. Matveev in [Ma] defined the following cardinal invariants (using different notation) p1 = min{κ : 2 is not strongly sequentially separable},