Combinatorics of Open Covers VI: Selectors for Sequences of Dense Sets

We consider the following two selection principles for topological spaces: Principle 1: For each sequence of dense subsets, there is a sequence of points from the space, the n-th point coming from the n-th dense set, such that this set of points is dense in the space; Principle 2: For each sequence of dense subsets, there is a sequence of finite sets, the n-th a subset of the n-th dense set, such that the union of these finite sets is dense in the space. We show that for separable metric space X one of these principles holds for the space Cp(X) of realvalued continuous functions equipped with the pointwise convergence topology if, and only if, a corresponding principle holds for a special family of open covers of X. An example is given to show that these equivalences do not hold in general for Tychonoff spaces. It is further shown that these two principles give characterizations for two popular cardinal numbers, and that these two principles are intimately related to an infinite game that was studied by Berner and Juhász. The following two selection hypotheses occur in many contexts in mathematics, especially in diagonalization arguments: Let N denote the set of positive integers and let A and B be collections of subsets of an infinite set. The hypothesis S1(A,B) states that for each sequence (On : n ∈ N) with terms in A there is a sequence (Tn : n ∈ N) such that for each n Tn ∈ On, and {Tn : n ∈ N} ∈ B. The hypothesis Sfin(A,B) states that for every sequence (On : n ∈ N) of elements of A there is a sequence (Tn : n ∈ N) such that for each n Tn is a finite subset of On, and ∪ ∞ n=1Tn is an element of B. A pair (A,B) for which either of these hypotheses holds usually has a rich theory. Consider the following game which is inspired by S1(A,B): Players ONE and TWO play an inning per n ∈ N. In the n-th inning ONE selects a set On ∈ A, after which TWO selects an element Tn ∈ On. A play (O1, T1, O2, T2, . . .) is won by TWO if {Tn : n ∈ N} is in B; otherwise ONE wins. Let G1(A,B) denote this game. The hypothesis H1(A,B) states that ONE has no winning strategy in G1(A,B). We have the implication H1(A,B) ⇒ S1(A,B). For several important examples of A and B it happens that the converse implication is also true. When this happens the game is a powerful tool to extract mathematical information about A and B. Subject Classification 90D44 Supported in part by NSF grant DMS 95-05375