Screenability and classical selection principles Liljana Babinkostova Boise State University

Types of open covers O : open covers of X Λ: An open cover C is a λ-cover if for each x in X the set {U  C: x  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 C Г: An open infinite cover C is a γ-cover if for each x in X, the set {U  C: x is not in U} is finite Г  Ω  Λ  O

S fin ( A, B ) For each sequence (U n :n  N) of elements of A, there is a sequence (V n : n  N) such that: K. Menger W. Hurewicz 1. for each n  N, V n is a finite subset of U n 2.  {V n : n  N} is an element of B. S1(A,B)S1(A,B) 1. for each n  N, V n  U n 2. {V n : n  N} is an element of B. Dr. Fritz Rothberger 14 October May 2000

Screenability For each open cover U of X there is a sequence (V n : n  N ) such that 1.Each V n is pairwise disjoint 2.Each V n refines U and 3.U{V n : n  N} is an open cover of X. R. H. Bing October 20, 1914–April 28, 1986

C-space (Selective screenability) For each sequence of open covers ( U n : n  N ) there is a sequence (V n : n  N ) such that: 1) Each V n is pairwise disjoint, 2) Each V n refines U n and 3) U {V n : n  N } is an open cover of X Dr. John Gresham Ranger College, TX D.F. Addis Texas Christian University

Selection principle S c ( A, B ) For each sequence ( U n : n  N ) of elements of A there is a sequence (V n : n  N ) such that 1.Each V n is pairwise disjoint and refines U n 2.U {V n : n  N} is an element of B

S c ( A, B ) - NEW selection property Baire space ╞ S c ( O, O ) + not S fin ( O, O ) Hilbert cube ╞ S fin ( O, O ) + not S c ( O, O )

Relationships S 1  A  B  S C ( A, B   S C ( A    S 1 ( A    ≠ S fin ( A    S C (   S 1 (   S fin (   S C (   S 1 (   S fin (  

S c and dimension theory Theorem (Addis and Gresham) : countable dimensional  S c (O,O)  weakly infinite dimensional.

The game G k c ( A, B ) 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 disjoint 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. NOTE: k is allowed to be any ordinal > 0.

Finite dimension Theorem: 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) but not in G n c (O,O).

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

Selective screenability  ??? ONE does not have a winning strategy

Theorem: If X is metric space that has S fin (O,O) the following are equivalent: 1.X has S c (O,O) 2.ONE does not have a winning strategy in G ω c (O,O).