1. Selection Principles and Basis Properties Liljana Babinkostova Boise State University

2. Menger Basis Property K. Menger (Property M,1924) : For each base B for the topology of the metric space (X,d), there is a sequence (B n : n <∞ ) such that for each n, B n  B and lim n → ∞ diam (B n )=0 and {B n : n <∞} covers X.

3. (Property E*, 1925): For each sequence (U n : n  N) of open covers, there is a sequence (V n : n  N) such that: 1. For each n  N, V n is a finite subset of U n 2.  {V n : n  N} is an open cover for X. S fin ( O, O ) W. Hurewicz O : the open covers of X

4. Relative version (2004): Let X be a metrizable space X and Y  X. The following statements are equivalent: 1.S fin (O X,O Y ) 2.For each base B of X there is a sequence (V n :n<∞) from the base such that {V n :n<∞} covers Y and lim n→∞ diam(V n )=0. (Hurewicz,1925): A metrizable space X has S fin (O,O) if and only if has property M with respect to each metric generating the topology of X. S fin (O,O) and Property M

5. S 1 (O,O) F. Rothberger Rothberger property (1937): A space X has property C`` (Rothberger property) if for each sequence of open covers (U n :n<∞) of X there is a sequence (V n :n<∞) such that for each n, V n  U n and {V n :n<∞} covers X.

6. Property M` Rothberger basis property (Sierpinski, 1937): A metrizable space X has property M` (Rothberger basis property) if for each base B for the topology of X and for each sequence (ε n :n<∞) of positive real numbers there is a sequence (B n :n<∞) such that for each n, B n  B and diam(B n )<ε n and {B n :n<∞} covers X. W.Sierpinski

7. Old results and two problems Property C``  Property M` Does M`  C`` How is M` related to C` Problem 1: Does M`  C`` ? Problem 2: How is M` related to C`= S 1 ( Fin.Op.Cov,O)? Fremlin and Miller, 1988: Property M does not imply property C`. Property C` does not imply property M. F. Rothberger, 1938:

8. Solution to Rothberger’s problems Let X be a metrizable space with S fin (O,O). The following are equivalent: 1. Y  X has the relative Rothberger basis property. 2. Y has the relative Rothberger property in X. Question 1: Does M`  C`` ? YES! Question 2: How is M` related to C`=S 1 ( Fin.Op.Cov,O)? M`  C`, but C`  M`

9. Selection Principle S c (O,O) For each sequence (U n : n  N ) of open covers of X 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 open cover for X. R. H. Bing

10. Basis Screenability property Metrizable space (X,d) has the Basis screenability property if for each basis B and for each sequence (ε n : n<∞) of positive real numbers there is a sequence (B n : n<∞) such that 1. For each n, B n  B is pairwise disjoint 2. For each n, and for each B  B n, diam(B)<ε n 3.  {B n :n<∞} covers X

11. For (X,d) a metric space with S fin (O,O) the following are equivalent: 1. X has the Basis Screenability property 2. S c (O,O)

12. Hurewicz covering property (1925): For each sequence (U n : n <∞) of open covers of X there is a sequence (V n : n <∞) such that: 1) For each n, V n  U n is finite and 2) For each y  Y for all but finitely many n, y   V n. (2001): For each base B of X there is a sequence (U n :n<∞) such that {U n :n<∞} is a groupable cover for Y and lim n→∞ diam(U n )=0. Hurewicz basis property Groupable: There is a partition U=  V n of U into finite sets V n such that for each m  n, V m  V n = , and for each x  X, for all but finitely many n, x  V n.