Transcript Selection Principles and Basis Properties

Selection Principles and Basis
Properties
Liljana Babinkostova
Boise State University
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 (Bn: n<∞) such that for
each n, Bn  B and limn→∞diam (Bn)=0 and {Bn: n <∞} covers X.
Sfin(O,O)
W. Hurewicz
(Property E*, 1925): For each sequence (Un: nN) of
open covers, there is a sequence (Vn: nN) such that:
1. For each n  N, Vn is a finite subset of Un
2.  {Vn : n N} is an open cover for X.
O : the open covers of X
Sfin(O,O) and Property M
(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.
Relative version
(2004): Let X be a metrizable space X and YX. 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.
S1(O,O)
F. Rothberger
(1937): A space X has property C`` (Rothberger 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`
W.Sierpinski
(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 (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.
Old results and two problems
F. Rothberger, 1938:
Property C`` Property M`
Problem 1: Does M`C`` ?
Problem 2: 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.
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.
2.
3.
For each n, Bn B is pairwise disjoint
For each n, and for each BBn, diam(B)<εn
{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
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 yY for all but finitely many n, y  Vn.
(1925):
Hurewicz basis property
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.
(2001): For
Groupable: There is a partition U=Vn of U into finite sets Vn such that for each mn,
VmVn=, and for each xX, for all but finitely many n, xVn.