Selection Principles and Basis Properties

Liljana BabinkostovaBoise State University

Menger Basis PropertyK. Menger

(Property M,1924) : For each base B for the topology of the

metric space (X,d), there is a sequence (Bn: n<∞) such that for

each n, Bn B and limn→∞diam (Bn)=0 and {Bn: n <∞} covers X.

(Property E*, 1925): For each sequence (Un: nN) ofopen covers, there is a sequence (Vn: nN) such that:

1. For each n N, Vn is a finite subset of Un 2. {Vn : n N} is an open cover for X.

Sfin(O,O)

W. Hurewicz

O : the open covers of X

Relative version

(2004): Let X be a metrizable space X and YX. The following statements are equivalent:

1.Sfin(OX,OY) 2.For each base B of X there is a sequence (Vn:n<∞) from the base such that {Vn:n<∞} covers Y and limn→∞diam(Vn)=0.

(Hurewicz,1925): A metrizable space X has Sfin(O,O) if and only if has property M with respect to each metric generating the topology of X.

Sfin(O,O) and Property M

S1(O,O)

F. Rothberger

(1937): A space X has property C`` (Rothberger propertyRothberger property) if for each sequence of open covers (Un:n<∞) of X there is a sequence (Vn:n<∞) such that for each n, Vn Un and {Vn:n<∞} covers X.

Property M`

(Sierpinski, 1937): A metrizable space X has property M` (Rothberger basis propertyRothberger basis property) if for each base B for thetopology of X and for each sequence (εn:n<∞) of positive real numbers there is a sequence (Bn:n<∞) such that for each n, Bn B and diam(Bn)<εn and {Bn:n<∞} covers X.

W.Sierpinski

Old results and two problems

Property C`` Property M`

Problem 1: Does M`Does M`C``C`` ?Problem 2: How is M` related to C`How is M` related to C`= S1(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:

Solution to Rothberger’s problems

Let X be a metrizable space with Sfin(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`=S1(Fin.Op.Cov,O)?

M` C`, but C` M`

Selection Principle Sc(O,O)

For each sequence (Un: n N ) of open covers of X

there is a sequence (Vn: n N ) such that

1. Each Vn is pairwise disjoint and refines Un

2. U {Vn: n N} is an open cover for X.

R. H. Bing

Basis Screenability property

Metrizable space (X,d) has the Basis screenability

property if for each basis B and for each sequence

(εn: n<∞) of positive real numbers there is a sequence

(Bn: n<∞) such that

1. For each n, Bn B is pairwise disjoint

2. For each n, and for each BBn, diam(B)<εn

3. {Bn:n<∞} covers X

For (X,d) a metric space with Sfin (O,O) the

following are equivalent:

1. X has the Basis Screenability property

2. Sc(O,O)

Hurewicz covering property

(1925): For each sequence (Un: n<∞) of open covers of X there is a sequence (Vn: n<∞) such that: 1) For each n, Vn Un is finite and 2) For each yY for all but finitely many n, y Vn.

(2001): For each base B of X there is a sequence (Un:n<∞) such that {Un:n<∞} is a groupable cover for Y and limn→∞diam(Un)=0.

Hurewicz basis property

Groupable: There is a partition U=Vn of U into finite sets Vn such that for each mn, VmVn=, and for each xX, for all but finitely many n, xVn.