Introduction 0-dimensional space X is 0-dimensional iff for each open cover U of X there is a disjoint refinement that is still an open cover of X.

Screenability (R.H.Bing) A topological space X is a screenable if: For each open cover U of X there is a sequence ( V n : n Є N ) such that 1. V n is pairwise disjoint family of open sets 2. For each n, V n U 2. For each n, V n refines U 3. U V n is a cover of X

C-space (W.Haver) A metric space X is a C-space if: For each sequence (  n : n Є N ) of positive real numbers there is a sequence (V n : n Є N ) such that 1) V n is a pairwise disjoint family of open sets 2) For each n, if V Є V n n 2) For each n, if V Є V n then diam(V) <  n U V n is a cover of X 3) U V n is a cover of X

C-space (D.F.Addis and G.J. Gresham) A topological space X is a C-space if: For each sequence of open covers ( U n : n Є N ) of X there is a sequence (V n : n Є N ) such that 1) V n is a pairwise disjoint family of open sets 2) For each n, V n U n and 2) For each n, V n refines U n and U V n is a cover of X 3) U V n is a cover of X

PART I Selection principle S c ( A, B )

Let A and B be collections of families of subsets of X. A topological space X has the S c ( A, B ) property if A topological space X has the S c ( A, B ) property if for each sequence ( U n : n Є N ) of elements of A there is a sequence (V n : n Є N ) such that 1) V n is a pairwise disjoint family of sets 2) For each n, V n U n and 2) For each n, V n refines U n and U V n is an element of B 3) U V n is an element of B

Types of open covers O - open cover Λ: An open cover C is a λ-cover if for each x in X the set {U Є C: x Є U} is infinite Ω: An open cover C is an ω-cover if each finite subset of X is a subset of some element of C and X doesn’t belong to any element of C {U Є C: x Є U} is infinite Ω: An open cover C is an ω-cover if each finite subset of X is a subset of some element of C and X doesn’t belong to any element of C Г: An open infinite cover C is a γ-cover if for each x in X, the set Г: An open infinite cover C is a γ-cover if for each x in X, the set {U Є C: x is not in U} is finite {U Є C: x is not in U} is finite Гsubset of Ω subset of Λ subset of O

S c ( A, B ) - NEW selection property Hilbert cube and Baire space S c ( A, B )≠ S fin ( A, B ) S fin ( A, B ): For each sequence (U n :n Є N) of elements in A, exists a sequence (V n :neN), such that for each n Є N, V n is a finite subset of U n and UV n is an element of B.

S c - property S c (O,O)=S c ( Λ, Λ )=S c ( Λ,O) S c (Ω,O)=S c (Ω, Λ ) S c (Ω, Λ )=S c ( Λ, Λ )

S c - property If the topological space satisfies S c (Ω, Ω), then for each m, X m satisfies S c (Ω, Ω). S c (O,O)->S c (Ω,O) S c (Г, Ω)->S c (Ω,O wgp ) S c (Г, Ω)->S c (Ω,O wgp )

Part 2 Games and covering dimension

The game G k c ( A, B ) The players play a predetermined number k of innings. The players play a predetermined number k of innings. In the n-th inning ONE chooses any O n from A, TWO responds with a refinement T n. In the n-th inning ONE chooses any O n from A, TWO responds with a refinement T n. A play ((O j,T j ): j< k) is won by TWO if U{T j : j < k } is in B ; else ONE wins. A play ((O j,T j ): j< k) is won by TWO if U{T j : j < k } is in B ; else ONE wins. NOTE: k is allowed to be any ordinal > 0. NOTE: k is allowed to be any ordinal > 0.

Finite dimension Theorem 1: For metrizable spaces X, for finite n the following are equivalent: 1. dim (X) = n. 2. TWO has a winning strategy in G n+1 c (O,O)

Countable dimension A metrizable space X is countable dimensional if it is a union of countably many zero-dimensional subsets. (Hurewicz, Wallman).

Alexandroff’s problem Addis and Gresham observed: countable dimensional -> S c (O,O) -> weakly infinite dimensional. Alexandroff’s problem: Does weakly infinite dimensional imply countable dimensional? Pol (1981): No. There is a compact metrizable counterexample.

Theorem 2: For metrizable space X the following are equivalent: 1) X is countable dimensional. 2) TWO has a winning strategy in G  c ( O, O ).

The Gc-type of a space. For any space there is an ordinal  such that TWO wins within  innings. gctp(X) = min{  :TWO wins in  innings}. Theorem: In any metrizable space X, gctp(X) ≤  1.

The Gc-type of a space. Theorem: For space X, if gctp(X) <  , then X has S c (O,O). Theorem: For X Pol ’ s counterexample for the Alexandroff problem, gctp(X)=  +1. Theorem: If X is metric space with S fin (O,O), the following are equivalent: 1) X has S c (O,O). 2) ONE has no winning strategy in G  c (O,O).

The Gc-type of a space. Pol’s example is S c (O,O). 1) Since it is compact, it is S fin (O,O), and so ONE has no winning strategy in G  c (O,O). 2) Since it is not countable dimensional, TWO has no winning strategy in G  c (O,O). Thus in Pol’s example the game G  c (O,O) is undetermined.