Topological Games and Covering Dimension

We consider a natural way of extending the Lebesgue covering dimension to various classes of infinite dimensional separable metric spaces. All spaces in this paper are assumed to be separable metric spaces. Infinite games can be used in a natural way to define ordinal valued functions on the class of separable metric spaces. One of our examples of such an ordinal function coincides in any finite dimensional metric spaces with the covering dimension of the space, and may thus be thought of as an extension of Lebesgue covering dimension to all separable metric spaces. We will call this particular extension of Lebesgue covering dimension the game dimension of a space. Game dimension is defined using a game motivated by a selection principle. Several natural classical selection principles are related to the one motivating game dimension, and their associated games can be used to compute upper bounds on game dimension. These games, and the upper bounds they provide, are interesting in their own right. We develop two such games and use them then to obtain upper bounds on our game dimension. We also compute the game dimension of a few specific examples. 1. Selection principles, open covers and games The selection principle S1(A,B) states that there is for each sequence (An : n ∈ N) of elements of A a sequence (Bn : n ∈ N) such that for each n we have Bn ∈ An, and {Bn : n ∈ N} ∈ B. The selection principle Sc(A,B) states that there is for each sequence (An : n ∈ N) of members of A a sequence (Bn : n ∈ N) of sets such that for each n, Bn is a pairwise disjoint family of sets and refines An and ⋃ n∈N Bn ∈ B. For a collection T of subsets of a topological space X we call an open cover U of X a “T -cover” if for each T ∈ T , there is a U ∈ U with T ⊆ U . The symbol O(T ) denotes the collection of T -covers of X . A trivial situation, but one we cannot ignore, arises when X itself is a member of T . We don’t follow the usual practice of requiring X 6∈ U for U ∈ O(T ). The motivation, as will be seen below, is that allowing this trivial situation provides a uniformity to the statements of some of our results. A few special combinatorial properties of the family T are important for our considerations. Here are some of them: T is said to be up-directed if for all A and B in T there is a C in T with A ⋃ B ⊂ C. It is said to be first-countable if there is for each T ∈ T a sequence (Un : n ∈ N) of open sets such that for each n, Un ⊃ Un+1 ⊃ T , and for each open set V ⊃ T , there is an n with V ⊇ Un. We shall say that X is T -first countable if there is for each T ∈ T a sequence (Un : n = 1, 2, . . .) of open sets such that for all n, T ⊂ Un+1 ⊂ Un, and for each open set U ⊃ T there is an n with Un ⊂ U . Let 〈T 〉 denote the subspaces which