Selective screenability, products and topological

groups Liljana BabinkostovaBoise State University

III Workshop on Coverings, Selections and Games in TopologyApril 25-29, 2007

Serbia

Selection principle Sc(A,B)

For each sequence (Un: n<∞) of elements of Athere is a sequence (Vn: n<∞) such that:

A and B are collections of families of subsets of an infinite set.

1) Each Vn is a pairwise disjoint family of sets,

2) each Vn refines Un and

3) {Vn: n<∞} is an element of B.

The game Gck(A,B)

The players play a predetermined ordinal number k of innings.

ONE chooses any On from A,

TWO responds with a disjoint refinement Tn of On.

TWO wins a play ((Oj,Tj): j< k) if {Tj : 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 Gcn+1(O,O),

but not in Gcn(O,O).

Gck(A,B) and Dimension

O denotes the collection of all open covers of X.

Gck(A,B) and Dimension

For metrizable spaces X the following are equivalent:

(i) X is countable dimensional.

(ii) TWO has a winning strategy in Gc(O,O).

HAVER PROPERTY

3) {Vn: n<∞} is an open cover of X.

W.E.Haver, A covering property for metric spaces, (1974)

For each sequence (εn: n<∞) of positive real numbers

there is a corresponding sequence (Vn:n<∞) where

1) each Vn is a pairwise disjoint family of open sets,

2) each element of each Vn is of diameter less than εn, and

Property C and Haver property

When A = B = O, the collection of open covers, Sc(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 Sc(O,O) implies the Haver property

2) countable dimension => Sc(O,O) => Sc(T,O)

Notes:

1) T is the collection of two-element open covers.

2) Sc(T,O) is Aleksandroff’s weak infinite dimensionality

Haver Property does not imply Sc(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 (Un: n<∞) of open covers of X

W. Hurewicz, Über eine Verallgemeinerung des Borelschen Theorems, (1925)

Hurewicz Property:

there is a sequence (Vn: n<∞) of finite sets such that

1) For each n, Vn Un and

2) each element of X is in all but finitely many of the sets Vn.

The Haver- and Hurewicz- properties

For X a metric space with the Hurewicz property,

the following are equivalent:

1)1) Sc(O,O) holds.

22)) 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 Sc(O,O)

(Hattori & Yamada; Rohm,1990) : Let X and Y be topological

spaces with Sc(O,O). If X is σ-compact, then XxY has

Sc(O,O).

(R.Pol,1995): (CH) For each positive integer n there is a separable metric space X such that 1) Xn has Sfin(O,O) and Sc(O,O), and

2) Xn+1 has Sfin(O,O), but not Sc(O,O).

Products and Sc(O,O)

Let X be a metric space which has Sc(O,O)

and Xn has the Hurewicz property.

Then Xn has property Sc(O,O).

If X and Y are metric spaces with Sc(O,O), and if XxY has the Hurewicz property then XxY has Sc(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 XxY has the

Haver property.

Let X and Y be metrizable spaces such that X has the Haver property and Y is countabledimensional. Then XxY has the Haver property.

Let X and Y be metrizable spaces with the Haver property. If X has the Hurewicz property then XxY has the Haver property.

Haver and Sc(O,O) in topological groupsLet (G,*) be a topological group and U be an

open nbd of the identity element 1G.

Open cover of G: Onbd(U)={x*U: xG}

Collection of all such open covers of G:

Onbd={Onbd(U): U nbd of 1G}

Haver and Sc(Onbd,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 Sc(Onbd, O).

Products and Sc(Onbd,O) in metrizable groups

Let (G,*) be a group which has property

Sc(Onbd,O) and the Hurewicz property.

Then if (H,*) has Sc(Onbd,O), GxH also has

Sc(Onbd,O).

Games and Sc(Onbd,O) in metrizable

groups

If (G,*) is a metrizable group then TFAE:

1. TWO has a winning strategy in Gc(Onbd, O).

2. G is countable dimensional.

Relation to Rothberger- and Menger-bounded groups

S1(Onbd,O) Sc(Onbd,O)

S1(nbd,O) Sc( 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

Ocfd and Ofd covers

Ocfd – all open covers U of X such that: X is not in U and for each CCFD there is a UU with C U .

Ofd – all open covers U of X such that:

X is not in U and for each CFD there is a UU with C U .

Selection principle S1(A,B)

A and B are collections of families of subset

of an infinite set.

For each sequence (Un: n<∞) of elements

of A there is a sequence (Vn: n<∞) such

that:

1) For each n, Vn Un

2) {Vn: nN } B.

Sc(O,O) and S1(Ofd,O)

Let X be a metrizable space.

S1(Ofd,O) => Sc(O,O)

Sc(O,O) ≠> S1(Ofd,O)

New classes of weakly infinite dimensional spaces

SCD CD

S1(Ocfd,O) S1(Ofd,O) Sc(O,O) Sc(T,O)

Sfin(O,O) Sfin(Ocfd,O) Sfin(Ofd,O)

KCD

S1(Okfd,O)

S1(Ofd,O) and products

If X has property S1(Ofd,O) and Y is

countable dimensional, then XxY has

property S1(Ofd,O).

S1(Ocfd,O) and products

If X has property S1(Ocfd,O) and Y is

strongly countable dimensional, then

XxY has property S1(Ocfd,O).

Thank you!

III Workshop on Coverings, Selections and Games in Topology

April 25-29, 2007Serbia