Selection principles and countable dimension

We consider player TWO of the game G1(A,B) when A and B are special classes of open covers of metrizable spaces. Our results give gametheoretic characterizations of the notions of a countable dimensional and of a strongly countable dimensional metric spaces. The selection principle S1(A,B) states: There is for each sequence (An : n ∈ N) of elements of A a corresponding sequence (bn : n ∈ N) such that for each n we have bn ∈ An, and {bn : n ∈ N} is an element of B. There are many examples of this selection principle in the literature. One of the earliest examples of it is known as the Rothberger property, S1(O,O). Here, O is the collection of all open covers of a topological space. The following game, G1(A,B), is naturally associated with S1(A,B): Players ONE and TWO play an inning per positive integer. In the n-th inning ONE first chooses an element On of A; TWO responds by choosing an element Tn ∈ On. A play O1, T1, O2, T2, · · · , On, Tn, · · · is won by TWO if {Tn : n ∈ N} is in B, else ONE wins. TWO has a winning strategy in G1(A,B) ⇓ ONE has no winning strategy in G1(A,B) ⇓ S1(A,B). There are many known examples of A and B where neither of these implications reverse. Several classes of open covers of spaces have been defined by the following schema: For a space X , and a collection T of subsets of X , an open cover U of X is said to be a T cover if X is not a member of U , but there is for each T ∈ T a U ∈ U with T ⊆ U . The symbol O(T ) denotes the collection of T -covers of X . In this paper we consider only A which are of the form O(T ) and B = O. Several examples of open covers of the form O(T ) appear in the literature. To mention just a few: When T is the family of one-element subsets of X , O(T ) = O. When T is the family of finite subsets of X , then members of O(T ) are called ω-covers in [3]. The symbol Ω denotes the family of ω-covers of