Selective screenability, products and topological groups Liljana Babinkostova Boise State University III Workshop on Coverings, Selections and Games in Topology April 25-29, 2007 Serbia

Selection principle S c ( A, B ) For each sequence ( U n : n<∞) of elements of A there is a sequence (V n : n<∞ ) such that: A and B are collections of families of subsets of an infinite set. 1) Each V n is a pairwise disjoint family of sets, 2) each V n refines U n and 3)  {V n : n<∞} is an element of B.

The game G c k ( A, B ) The players play a predetermined ordinal number k of innings. ONE chooses any O n from A, TWO responds with a disjoint refinement T n of O n. TWO wins a play ((O j,T j ): j< k) if  {T j : j < k } is in B ; else ONE wins. In inning n:

For metrizable spaces X, for finite n the following are equivalent: (i) dim (X) = n. (ii) TWO has a winning strategy in G c n+1 (O,O), but not in G c n (O,O). G c k ( A, B ) and Dimension O denotes the collection of all open covers of X.

G c k ( A, B ) and Dimension For metrizable spaces X the following are equivalent: (i) X is countable dimensional. (ii) TWO has a winning strategy in G c  (O,O).

HAVER PROPERTY 3)  {V n : n<∞ } is an open cover of X. W.E.Haver, A covering property for metric spaces, (1974) For each sequence (ε n : n<∞) of positive real numbers there is a corresponding sequence (V n :n<∞) where 1) each V n is a pairwise disjoint family of open sets, 2) each element of each V n is of diameter less than ε n, and

Property C and Haver property When A = B = O, the collection of open covers, S c (O,O) is also known as property C. D. Addis and J. Gresham, A class of infinite dimensional spaces I, (1978) 1) In metric spaces S c (O,O) implies the Haver property 2) countable dimension => S c (O,O) => S c (T,O) Notes: 1)T is the collection of two-element open covers. 2)S c (T,O) is Aleksandroff’s weak infinite dimensionality

Haver Property does not imply S c (O,O) Example X=M  L M - complete metric space, totally disconnected strongly infinite dimensional. L – countable dimensional s.t. M  L is compact metric space.

Z L – countable dimensional Haver property ≠ Property C L M Z  M, Z – zero dimensional (compact subset of totally disconnected space) Z

Alexandroff ’ s Problem Is countable dimensionality equivalent to weak infinite dimensionality? R.Pol (1981): No. There is a compact metrizable counterexample.

The Hurewicz property For each sequence (U n : n <∞ ) of open covers of X W. Hurewicz, Ü ber eine Verallgemeinerung des Borelschen Theorems, (1925) Hurewicz Property: there is a sequence (V n : n<∞) of finite sets such that 1) For each n, V n  U n and 2) each element of X is in all but finitely many of the sets  V n.

The Haver- and Hurewicz- properties For X a metric space with the Hurewicz property, the following are equivalent: 1) 1) S c (O,O) holds. 2) 2) X has the Haver property in some equivalent metric on X. 3) 3) X has the Haver property in all equivalent metrics on X.

Products and S c (O,O) (Hattori & Yamada; Rohm,1990) : Let X and Y be topological spaces with S c (O,O). If X is σ-compact, then X x Y has S c (O,O). (R.Pol,1995): (CH) For each positive integer n there is a separable metric space X such that 1) X n has S fin (O,O) and S c (O,O), and 2) X n+1 has S fin (O,O), but not S c (O,O).

Products and S c (O,O) Let X be a metric space which has S c (O,O) and X n has the Hurewicz property. Then X n has property S c (O,O). If X and Y are metric spaces with S c (O,O), and if X x Y has the Hurewicz property then X x Y has S c (O,O).

Products and the Haver property Let X be a complete metric space which has the Haver property. Then for every metrizable space Y which has the Haver property, also X x Y has the Haver property. Let X and Y be metrizable spaces such that X has the Haver property and Y is countable dimensional. Then X x Y has the Haver property. Let X and Y be metrizable spaces with the Haver property. If X has the Hurewicz property then X x Y has the Haver property.

Haver and S c (O,O) in topological groups Let (G,*) be a topological group and U be an open nbd of the identity element 1 G. Open cover of G: O nbd (U)={x * U: x  G} Collection of all such open covers of G: O nbd ={Onbd(U): U nbd of 1 G }

Haver and S c (O nbd,O) in topological groups Let (G, * ) be a metrizable topological group. The following are equivalent: (i) G has the Haver property in all left invariant metrics. (ii) G has the property S c (O nbd, O).

Products and S c (O nbd,O) in metrizable groups Let (G, * ) be a group which has property S c (O nbd,O) and the Hurewicz property. Then if (H, * ) has S c (O nbd,O), G x H also has S c (O nbd,O).

Games and S c (O nbd,O) in metrizable groups If (G, * ) is a metrizable group then TFAE: 1. TWO has a winning strategy in G  c (O nbd, O). 2. G is countable dimensional.

Relation to Rothberger- and Menger-bounded groups S 1 (O nbd,O) S c (O nbd,O) S 1 (  nbd,O)S c (  nbd,O) None of these implications reverse.

New classes of open covers X-separable metric space CFD: collection of closed, finite dimensional subsets of X FD: collection of all finite dimensional subsets of X

O cfd and O fd covers O cfd – all open covers U of X such that: X is not in U and for each C  CFD there is a U  U with C  U. O f d – all open covers U of X such that: X is not in U and for each C  FD there is a U  U with C  U.

Selection principle S 1 ( A, B ) A and B are collections of families of subset of an infinite set. For each sequence (U n : n <∞ ) of elements of A there is a sequence (V n : n <∞ ) such that: 1) For each n, V n  U n 2) {V n : n  N }  B.

S c (O,O) and S 1 (O fd, O) Let X be a metrizable space. S 1 (O fd, O) => S c (O,O) S c (O,O) ≠ > S 1 (O fd, O)

New classes of weakly infinite dimensional spaces SCDCD S 1 (O cfd,O)S 1 (O fd,O)S c (O,O)S c (T,O) S fin (O,O)S fin (O cfd,O)S fin (O fd,O) KCD S 1 (O kfd,O)

S 1 (O fd, O) and products If X has property S 1 (O fd,O) and Y is countable dimensional, then X x Y has property S 1 (O fd,O).

S 1 (O cfd, O) and products If X has property S 1 (O cfd,O) and Y is strongly countable dimensional, then X x Y has property S 1 (O cfd,O).

Thank you! III Workshop on Coverings, Selections and Games in Topology April 25-29, 2007 Serbia