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Introduction
0-dimensional space
X is 0-dimensional iff for each open cover
U of X there is a disjoint refinement that is
still an open cover of X.
Screenability (R.H.Bing)
A topological space X is a screenable if:
For each open cover U of X there is a
sequence (Vn: nЄN ) such that
1.
2.
3.
Vn is pairwise disjoint family of open sets
For each n,Vn refines U
UVn is a cover of X
C-space (W.Haver)
A metric space X is a C-space if:
For each sequence (en: nЄN ) of positive real
numbers there is a sequence (Vn: nЄN )
such that
1) Vn is a pairwise disjoint family of open sets
2) For each n, if VЄVn then diam(V) < en
3) U Vn is a cover of X
C-space (D.F.Addis and G.J. Gresham)
A topological space X is a C-space if: For each
sequence of open covers (Un: nЄN ) of X there is
a sequence (Vn:nЄN ) such that
Vn is a pairwise disjoint family of open sets
2) For each n, Vn refines Un and
3) U Vn is a cover of X
1)
PART I
Selection principle Sc (A,B)
Selection principle Sc (A,B)
Let A and B be collections of families of subsets of X.
A topological space X has the Sc(A,B) property if for
each sequence (Un:
nЄN
) of elements of A there is
a sequence (Vn: nЄN ) such that
1)
2)
Vn is a pairwise disjoint family of sets
For each n, Vn refines Un and
3) U
Vn is an element of B
Types of open covers
O - open cover
Λ: An open cover C is a λ-cover if for each x in X the set
{UЄ C: x Є U} is infinite
Ω: An open cover C is an ω-cover if each finite subset of X is a subset
of some element of C and X doesn’t belong to any element of C
Г: An open infinite cover C is a γ-cover if for each x in X, the set
{U Є C: x is not in U} is finite
Гsubset of Ω subset of Λ subset of O
Sc(A,B) - NEW selection property
Hilbert cube and Baire space
Sc(A,B)≠ Sfin(A,B)
Sfin(A,B): For each sequence (Un:nЄN) of elements in
A, exists a sequence (Vn:neN), such that for each
nЄN, Vn is a finite subset of Un and UVn is an element
of B.
Sc - property
Sc(O,O)=Sc(Λ, Λ)=Sc(Λ,O)
Sc(Ω,O)=Sc(Ω,Λ)
Sc(Ω, Λ)=Sc(Λ,Λ)
Sc - property
If the topological space satisfies Sc(Ω, Ω),
then for each m, Xm satisfies Sc(Ω, Ω).
Sc(O,O)->Sc(Ω,O)
Sc(Г, Ω)->Sc(Ω,Owgp)
Part 2
Games and covering dimension
The game Gkc(A,B)



The players play a predetermined number
k of innings.
In the n-th inning ONE chooses any On
from A, TWO responds with a refinement
Tn.
A play ((Oj,Tj): j< k) is won by TWO if
U{Tj : j < k } is in B; else ONE wins.
NOTE: k is allowed to be any ordinal > 0.
Finite dimension
Theorem 1: For metrizable spaces X, for
finite n the following are equivalent:
1. dim (X) = n.
2. TWO has a winning strategy in Gn+1c(O,O)
Countable dimension
A metrizable space X is
countable dimensional
if it is a union of countably many
zero-dimensional subsets. (Hurewicz,
Wallman).
Alexandroff’s problem
Addis and Gresham observed:
countable dimensional -> Sc(O,O) -> weakly infinite dimensional.
Alexandroff’s problem: Does weakly infinite
dimensional imply countable dimensional?
Pol (1981): No. There is a compact metrizable
counterexample.
Theorem 2: For metrizable space X the
following are equivalent:
1) X is countable dimensional.
2) TWO has a winning strategy in Gwc(O,O).
The Gc-type of a space.
For any space there is an ordinal a such
that TWO wins within a innings.
gctp(X) = min{a :TWO wins in a innings}.
Theorem: In any metrizable space X,
gctp(X) ≤ w1.
The Gc-type of a space.
Theorem: For space X, if gctp(X) < w1, then X has
Sc(O,O).
Theorem: For X Pol’s counterexample for the
Alexandroff problem, gctp(X)= w +1.
Theorem: If X is metric space with Sfin(O,O), the
following are equivalent:
1) X has Sc(O,O).
2) ONE has no winning strategy in Gwc(O,O).
The Gc-type of a space.
Pol’s example is Sc(O,O).
1) Since it is compact, it is Sfin(O,O), and
so ONE has no winning strategy in
Gwc(O,O).
2) Since it is not countable dimensional,
TWO has no winning strategy in Gwc(O,O).
Thus in Pol’s example the game Gwc(O,O) is
undetermined.