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% The combinatorics of open covers (II)
% Winfried Just
% Arnold W. Miller
% Marion Scheepers
% Paul J. Szeptycki
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\newcommand{\zz}{{\Bbb Z}^\omega} \newcommand{\ufin}{{\sf U}_{fin}}
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\newcommand{\lm}{\Lambda} \newcommand{\ga}{\Gamma}
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\newcommand{\split}{{\sf Split}} \newcommand{\covmeag}{{\sf cov}({\cal M})}
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\begin{document}
\begin{center}
{\large \bf The combinatorics of open covers (II)}
\end{center}
\bigskip
\begin{center}
Winfried Just\footnote{partially supported by NSF grant
DMS-9312363},
Arnold W. Miller,
Marion Scheepers\footnote{partially supported by NSF grant DMS-95-05375},\\
and Paul J. Szeptycki
\end{center}
\bigskip
\begin{abstract} We continue to investigate various
diagonalization properties
for sequences of open covers of separable metrizable spaces
introduced in Part I. These properties generalize
classical ones of Rothberger, Menger, Hurewicz, and
Gerlits-Nagy. In particular, we show that most of the properties
introduced in Part I are indeed distinct. We characterize
two of the new properties by showing that they are equivalent
to saying all finite powers have one of the classical properties
above (Rothberger property in one case and in the Menger property
in other). We consider for each property the smallest
cardinality of metric space which fails to have that property.
In each case this cardinal turns out to equal another well-known
cardinal less than the continuum.
We also disprove (in ZFC) a conjecture of Hurewicz
which is analogous to the Borel conjecture. Finally,
we answer several questions from Part I concerning partition
properties of covers.
\footnote{
AMS Classification: 03E05 04A20 54D20
$\;$ Key words: Rothberger property $C^{\prime\prime}$,
Gerlits-Nagy property
$\gamma$-sets, Hurewicz property,
Menger property, $\gamma$--cover, $\omega$--cover,
Sierpi\'nski set, Lusin set.
Appears in Topology and its applications, 73(1996), 241-266.
}
\end{abstract}
\begin{center}{\bf Introduction}
\end{center}
Many topological properties of spaces have been defined or characterized
in terms of the properties of open coverings of these spaces.
Popular among such properties are the properties
introduced by Gerlits and Nagy \cite{G-N}, Hurewicz
\cite{Hu}, Menger \cite{Me} and Rothberger \cite{Ro}.
These are all defined in terms of the possibility of extracting
from a given sequence of open covers of some sort, an open cover of some
(possibly different) sort.
In Scheepers \cite{S} it was shown that
when one systematically studies the definitions involved and inquires whether
other natural variations of the defining procedures produce any new classes
of sets which have mathematically interesting properties, an aesthetically
pleasing picture emerges. In \cite{S} the basic implications were
established. It was left open whether these were the only implications.
Let $X$ be a topological space. By a ``cover'' for $X$ we always
mean ``countable open cover''. Since we are primarily interested in
separable metrizable (and hence Lindel\"of) spaces, the restriction
to countable covers does not lead to a loss of generality.
A cover ${\cal U}$ of $X$ is said to be
\begin{enumerate}
\item{{\em large} if for each $x$ in $X$ the set
$\{U\in {\cal U}:x\in U\}$ is infinite;}
\item{{\em an $\omega$--cover} if $X$ is not in ${\cal U}$ and for
each finite subset $F$ of $X$, there is
a set $U\in{\cal U}$ such that $F\subset U$;}
\item{{\em a $\gamma$--cover} if it is infinite and for each $x$ in
$X$ the set $\{U\in {\cal U}:x\not\in U\}$ is finite.}
\end{enumerate}
We shall use the symbols $\op$, $\lm$, $\om$ and
$\ga$ to denote the collections of all open, large, $\omega$ and
$\gamma$--covers respectively, of $X$. Let ${\cal A}$ and ${\cal B}$
each be one of these four classes. We consider the
following three
``procedures'', $\sone$, $\sfin$ and $\ufin$, for obtaining covers in
${\cal B}$ from covers in ${\cal A}$:
\begin{enumerate}
\item{$\sone({\cal A},{\cal B})$: For a sequence
$({\cal U}_n:n=1,2,3,\dots)$ of
elements of ${\cal A}$, select for each $n$ a set
$U_n\in{\cal U}_n$ such that
$\{U_n:n=1,2,3,\dots\}$ is a member of ${\cal B}$;}
\item{$\sfin({\cal A},{\cal B})$: For a sequence
$({\cal U}_n:n=1,2,3,\dots)$ of
elements of ${\cal A}$, select for each $n$ a finite set
${\cal V}_n\subset{\cal U}_n$ such that
$\cup_{n=1}^{\infty}{\cal V}_n$ is an element of ${\cal B}$;}
\item{$\ufin({\cal A},{\cal B})$: For a sequence
$({\cal U}_n:n=1,2,3,\dots)$ of elements of ${\cal A}$,
select for each $n$ a finite set ${\cal V}_n\subset{\cal U}_n$
such that
$\{\cup{\cal V}_n:n=1,2,3,\dots\}$ is a member of ${\cal B}$
or\footnote{This is similar to the $*$ convention of \cite{S}.}
there exists an $n$ such that $\cup{\cal V}_n=X$.}
\end{enumerate}
For ${\sf G}$ one of these three procedures, let us say that
a space has property
${\sf G}({\cal A},{\cal B})$ if for every sequence of elements
of ${\cal A}$, one can
obtain an element of ${\cal B}$ by means of procedure ${\sf G}$.
Letting ${\cal A}$ and
${\cal B}$ range over the set $\{\op, \lm, \om, \ga \}$, we see that
for each ${\sf G}$ there are potentially sixteen classes of
spaces of the form ${\sf G}({\cal A},{\cal B})$.
Each of our properties is monotone decreasing in the first coordinate
and increasing in the second, hence we get the following diagram
(see figure \ref{sfinfig})
for ${\sf G}=\sfin$.
\begin{figure}
\unitlength=1.00mm
\begin{picture}(80.00,80.00)(-12,30)
%\put(-5,30){\framebox(120,80){}}
\multiput(19,40)(0,20){4}{\multiput(0,0)(30,0){3}{\vector(1,0){12}}}
\multiput(10,45)(30,0){4}{\multiput(0,0)(0,20){3}{\vector(0,1){12}}}
\multiput(-5,70)(4,0){15}{\line(1,0){3}}
\multiput(57,50)(4,0){7}{\line(1,0){3}}
\multiput(55,70)(0,-4){5}{\line(0,-1){3}}
\multiput(85,50)(0,-4){4}{\line(0,-1){3}}
\put(10.00,40.00){\makebox(0,0)[cc]{$\sfin(\op,\ga)$}}
\put(40.00,40.00){\makebox(0,0)[cc]{$\sfin(\op,\om)$}}
\put(70.00,40.00){\makebox(0,0)[cc]{$\sfin(\op,\lm)$}}
\put(100.00,40.00){\makebox(0,0)[cc]{$\sfin(\op,\op)$}}
\put(10.00,60.00){\makebox(0,0)[cc]{$\sfin(\lm,\ga)$}}
\put(40.00,60.00){\makebox(0,0)[cc]{$\sfin(\lm,\om)$}}
\put(70.00,60.00){\makebox(0,0)[cc]{$\sfin(\lm,\lm)$}}
\put(100.00,60.00){\makebox(0,0)[cc]{$\sfin(\lm,\op)$}}
\put(10.00,80.00){\makebox(0,0)[cc]{$\sfin(\om,\ga)$}}
\put(40.00,80.00){\makebox(0,0)[cc]{$\sfin(\om,\om)$}}
\put(70.00,80.00){\makebox(0,0)[cc]{$\sfin(\om,\lm)$}}
\put(100.00,80.00){\makebox(0,0)[cc]{$\sfin(\om,\op)$}}
\put(10.00,100.00){\makebox(0,0)[cc]{$\sfin(\ga,\ga)$}}
\put(40.00,100.00){\makebox(0,0)[cc]{$\sfin(\ga,\om)$}}
\put(70.00,100.00){\makebox(0,0)[cc]{$\sfin(\ga,\lm)$}}
\put(100.00,100.00){\makebox(0,0)[cc]{$\sfin(\ga,\op)$}}
\end{picture}
\caption{Basic diagram for $\sfin$ \label{sfinfig}}
\end{figure}
It also
is easily checked that $\sfin(\lm,\om)$ and
$\sfin(\op,\lm)$ are impossible for nontrivial
$X$. Hence the five classes in the lower left corner
are eliminated. The same follows for the stronger
property $\sone$. In the case of $\ufin$ note that
for any class of covers {\cal B},
$\ufin(\op,{\cal B})$ is equivalent to $\ufin(\ga,{\cal B})$
because given an open cover $\{U_n: n\in\omega\}$ we may replace
it by the $\gamma$-cover, $\{\cup_{iN_i$. But then $x$ is in
$V^n_m$ for all
$m> \max\{N_i:i=1,\dots,n\}$.
Now apply $\sfin(\ga,\ga)$ to $({\cal V}_n :
n=1,2,\dots)$. We get a sequence
$$({\cal W}_n:n\in\omega)$$
such that ${\cal W}_n$ is a finite subset
of ${\cal V}_n$ for each $n$, and such that
$\cup_{n=1}^{\infty}{\cal W}_n$ is
a $\gamma$--cover of $X$.
Choose a an increasing sequence $n_12\cdot\max\{ k:U_{n,k}\in {\cal V}_n\}$$
for all $n\in \omega$.
Let $f,g\in { L}$ be such that $h=f+g$. Then
$$\max\{|f(n)|,|g(n)|\}\geq {1\over 2} h(n)$$
for all $n\in \omega$, and hence $\{f,g\}\not\subseteq \cup{\cal V}_n$
for any $n\in \omega$.
\qed
\newpage
\subsec{The special Sierpi\'nski set S}
A Sierpi\'nski set is an uncountable subset of the real line which
has countable intersection with every set of Lebesgue measure zero.
In Theorem 7 of Fremlin and Miller \cite{F-M} it was shown that
every Sierpi\'nski set
belongs to the class $\ufin(\ga,\ga)$. Sets with the property that every Borel image in the
Baire space is bounded were called $A_2$--sets in Bartoszynski
and Scheepers \cite{B-S}.
\begin{theorem}\label{sierpinskianda2} Every Sierpi\'nski set is an
$A_2$--set.
\end{theorem}
\pf
Let $X$ be a subset of the unit interval, and assume that $X$ is
a Sierpinski set. We may assume that $X$ has outer measure one
(else, replace it with the set of points in the unit interval which
are rational translations of elements of $X$). Let a Borel function
$\Psi$ from $X$ to $^{\omega}\omega$ be given. Extend it a a Borel
function $\Gamma$ from the unit interval to $^{\omega}\omega$.
For each $m$ and $n$, define
$S^m_n = \{x:\Gamma(x)(j) < n \mbox{ whenever }j\leq m\}$.
Then each $S^m_n$ is a Borel set and thus Lebesgue measurable.
Moreover, for each $m$, if $j i,\; F\subseteq U_{n}^{i}\}.
$$
As each ${\cal U}_{n}$ is a $\gamma$-cover, if $i>f_{F}(n)$, then
$F\subseteq U_{n}^{i}$. Therefore, $$\{f_{F}:F\in[X]^{<\omega}\}$$
must be a dominating family. Otherwise there is a $g$ not dominated
by any such $f_{F}$. I.e., for each finite $F\subseteq X$, there is
an integer $n$ such that $g(n)>f_{F}(n)$. This implies that
$\{U_{n}^{g(n)}:n\in \omega\}$ is an $\omega$-cover,
contradicting the failure of $S_{1}(\ga ,\om)$.
So $\non(\sone(\ga,\om))\geq {\goth d}$.
\qed
\begin{theorem}\label{sfinomega}
$\non(\sfin(\om,\om))={\goth d}$.
\end{theorem}
\pf Identical to the proof of \ref{gam4}. One only needs to modify
the definition of $f_F$ to
$$f_{F}(n)=min\{i:F\subseteq U_{n}^{i}\}$$
and take ${\cal V}_n=\{U^i_n:i\leq g(n)\}$.
\qed
\begin{theorem}\label{gam3} $\non(\sone(\ga,\ga))={\goth b}$.
\end{theorem}
\pf
Using $\sone(\ga,\ga)\subseteq \ufin(\ga,\ga)$ and Theorem~\ref{gam6}
it follows that $$\non(\sone(\ga,\ga))\leq {\goth b}.$$
Conversely, suppose that $X$ is a set of reals and that
$({\cal U}_{n})_{n\in \omega}$ is a sequence of $\gamma $-covers
witnessing the failure of $S_{1}(\ga ,\ga )$. For each
$x\in X$ define $f_{x}\in \oo$ by
$$f_{x}(n)=min\{i:\forall j\geq i, x\in U_{n}^{j}\}.$$
If $g$ were to dominate each $f_{x}$,
then $(U_{n}^{g(n)})_{n\in \omega}$ would be a $\gamma$-cover, a
contradiction. Therefore $\{f_{x}:x\in X\}$ is an unbounded
family. Hence $\non(\sone(\ga,\ga))\geq {\goth b}$.
\qed
\begin{theorem}\label{gam1}
$\non(\sone(\om,\om))=\covmeag$.
\end{theorem}
\pf
The inclusion $\sone(\om,\om)\subseteq \sone(\op,\op)$ and
Theorem~\ref{gam2} give us the inequality
$\non(\sone(\om,\om))\leq\covmeag$.
Conversely fix $X$ a set of reals and
$({\cal U}_{n})_{n\in \omega}$
a sequence of $\omega$-covers witnessing
the failure of $\sone(\om,\om)$. For each finite
$F\subseteq X$ let
$$
K_{F}=\{f\in \oo :(\forall n\in \omega)(F\not\subset U_{n}^{f(n)})\}.
$$
Since for each $f\in \oo$ there is a finite
$F\subseteq X$ such that $F\not\subset U_{n}^{f(n)}$, we have that
$\oo=\bigcup\{K_{F}:F\in [X]^{<\omega}\}$.
Furthermore, each $K_{F}$ is closed and
nowhere dense. Hence $\non(\sone(\om,\om))\geq\covmeag$.
\qed
Our results are summarized in figure \ref{cardfig}.
Classical results about the relationships between the cardinals
${\goth p}$, ${\goth b}$, ${\goth d}$ and $cov({\cal M})$ give alternative
proofs that many of the implications in our diagram cannot be
reversed.
\subsec{$\split(\lm,\lm)$ and $\split(\om,\om)$}
These properties were defined in
Scheepers \cite{S}: for classes of covers ${\cal A}$ and
${\cal B}$, a space has
property $\split({\cal A},{\cal B})$ iff every open cover
${\cal U\in A}$ can be partitioned into two subcovers ${\cal U}_{0}$ and
${\cal U}_{1}$ both in ${\cal B}$.
Recall that a family ${\cal R}\subseteq [\omega]^{\omega}$
is said to be a {\em reaping family} if for
each $x\in [\omega]^{\omega}$ there is a $y\in {\cal R}$ such
that either $y\subseteq^{*}x$ or $y\subseteq^{*} \omega\setminus x$.
The minimal cardinality of a reaping family is denoted by
${\goth r}$, and the minimal cardinality of a base for a nonprincipal
ultrafilter is denoted by ${\goth u}$. In the proofs of the next two theorems we will use the families $${\cal U}=\{B^{1}_{n}:n\in \omega\} \mbox{ and } {\cal V}=\{B^{0}_{n}:n\in \omega\}$$
where
$$B^{1}_{n}=\{x\in [\omega]^{\omega}:n\in x\}
\mbox{ and } B^{0}_{n}=\{x\in [\omega]^{\omega}:n\not\in x\}.$$
Note that ${\cal U}$ and ${\cal V}$ are large covers of any subset
of $[\omega]^{\omega}$ and ${\cal U}\cup{\cal V}$ is a subbase for the topology. We will refer to ${\cal U}$ as the
canonical large cover.
\begin{theorem}\label{large1}
$\non(\split(\lm,\lm))={\goth r}$.
\end{theorem}
\pf Suppose that $X\subseteq[\omega]^{\omega}$ is a
reaping family. Therefore the cover ${\cal U}$
cannot be partitioned into two large subcovers.
Conversely, suppose that $X$ is a set of reals and
$\{U_{n}:n\in \omega\}$ is a large cover of $X$.
For each $x\in X$ let
$$
A_{x}=\{n\in \omega:x\in U_{n}\}.
$$
If ${\cal F}$ is the collection of all such $A_{x}$'s, then
${\cal F}$ is a reaping family. For if $A\subseteq \omega$ is such that
for all $x\in X$ both $A_{x}\cap A$ and $A_{x}\setminus A$ are
infinite, then
$$\{U_{n}:n\in A\}\cup\{U_{n}:n\not\in A\}$$
is a splitting of $\{U_n:n\in \omega\}$ into disjoint large subcovers.
\qed
The proof yields a bit more.
\begin{theorem}\label{large2} A set of reals $X$ is $\split (\lm ,\lm )$
with respect to clopen covers if and only if every
continuous image of $X$ in $[\omega]^{\omega}$ is not a reaping
family.
\end{theorem}
\pf Suppose that $X$ is a set of reals,
$f:X\rightarrow [\omega]^{\omega}$ is continuous and that $f(X)$
is a reaping family. The canonical large cover is in fact a clopen
family. Therefore the collection $f^{-1}({\cal
U})=\{f^{-1}(B^{1}_{n}):n\in \omega\}$ is a large clopen cover of
$X$. Suppose $f^{-1}({\cal U})={\cal V}_{0}\cup{\cal V}_{1}$ is a
partition. Then we have the corresponding partition of
$\omega=A_{0}\cup A_{1}$ where
${\cal V}_{i}=\{f^{-1}(U_{n}):n\in A_{i}\}$.
As $f(X)$ is a reaping family, there is an $x\in X$ such
that for either $i=0$ or $1$, $f(x)\subseteq^{*} A_{i}$. Then
${\cal V}_{i}$ is not large at $x$.
Therefore $X$ is not $\split (\lm ,\lm )$ with respect to
the clopen cover $f^{-1}({\cal U})$.
\smallskip
\noindent Conversely, suppose that $X$ is not
$\split(\lm,\lm ) $ with respect to some large clopen cover
$\{U_{n}:n\in \omega\}$. For each $x\in X$ define
$f_{x}\in [\omega]^{\omega}$
by $n\in f_{x}$ iff $x\in U_{n}$. Since the cover is large, each $f_{x}$
is infinite. As above, since $\{U_n:n\in \omega\}$ cannot be split,
$\{f_{x}:x\in X\}$ is a reaping family. Therefore it suffices to
check that the mapping $f:x\rightarrow f_{x}$ is continuous. But
the collection of $\{B_n^i:n\in \omega,i=0,1\}$ forms a subbase
for $[\omega]^{\omega}$,
and clearly $f^{-1}(B_{n}^{1})=U_{n}$ and
$f^{-1}(B_{n}^{0})=X\setminus
U_{n}$ therefore $f$ is continuous (this is the only place where we
need the restriction to clopen covers).
\qed
\begin{theorem}\label{omega} $\non(\split(\om,\om))={\goth u}$.
\end{theorem}
\pf Suppose that $X\subseteq[\omega]^{\omega}$ is a filter-base.
Then the canonical large cover in $[\omega]^{\omega}$
is in fact an $\omega$-cover of $X$. If $X$ is a base for an
ultrafilter, then this cover cannot be partitioned into two $\omega$-subcovers.
\smallskip
\noindent Conversely, suppose that $X$ is a set of reals and ${\cal W}$
is an $\omega$-cover of $X$. For each $x\in X$ let
$$
{\cal W}_{x}=\{U\in {\cal W}:x\in U\}.
$$
If ${\cal F}$ is the collection of all such ${\cal W}_{x}$'s, then
${\cal F}$ forms a filterbase
on ${\cal W}$ and if ${\cal W}$ cannot be split into two
$\omega$-covers, then ${\cal F}$ generates a nonprincipal ultrafilter.
\qed
Analogously to Theorem \ref{large2} we can prove:
\begin{theorem}\label{omega2} A set of reals $X$ is
$\split(\om,\om)$ with respect to clopen covers if and only if every
continuous image of $X$ in $[\omega]^{\omega}$ does not
generate an ultrafilter.
\end{theorem}
Note that a base for an ultrafilter is a reaping family, and
therefore ${\goth r}\leq{\goth u}$. In \cite {belku}
it is proven consistent that
this inequality may be strict. Therefore
$\split(\lm,\lm )\not\Rightarrow \split (\om ,\om )$. Similarly
neither ${\goth r}$ nor ${\goth u}$ are comparable to ${\goth d}$,
therefore there are no implications between either
$\split(\lm,\lm )$ or $\split(\om,\om)$ and any of
the six classes in figure \ref{cardfig} whose `$\non$' is
equivalent to ${\goth d}$.
In Scheepers \cite{S} it is shown that
\begin{itemize}
\item $\ufin(\ga,\ga)\Rightarrow \split(\lm,\lm)$ (Cor 29), and
\item $\sone(\op,\op)\Rightarrow \split(\lm,\lm)$ (Thm 15).
\end{itemize}
Note that while both ${\goth b}\leq {\goth r}$
and $\covmeag\leq {\goth r}$, it is consistent that these
inequalities are strict (see Vaughan \cite{vaughan}).
So neither of these implications can be reversed.
\begin{problem}
Does $\split(\om,\om)\Rightarrow \split(\lm,\lm)$?
\end{problem}
\section{The Hurewicz Conjecture and the Borel Conjecture.}\label{hur}
Every $\sigma$--compact space belongs to $\ufin(\ga,\ga)$. It is
also well-known that not every space belonging to $\ufin(\ga,\ga)$ need
be $\sigma$--compact. We now look at the traditional examples of sets
of reals belonging
to $\ufin(\ga,\ga)$, and show that some of these belong to
$\sone(\ga,\ga)$, while others do not. Since $\sone(\ga,\ga)$ is
contained in $\sone(\ga,\lm)$,
and the unit interval is not an element of $\sone(\ga,\lm)$, we see that
the $\sigma$--compact spaces do not in general belong to the class
$\sone(\ga,\ga)$.
On page 200 of \cite{Hu}, W. Hurewicz conjectures:
\medskip
\noindent
{\em
[Hurewicz] A set of real numbers has property
$\ufin(\ga,\ga)$ if, and only if, it is
$\sigma$--compact.\footnote{``Es
entsteht nun die Vermutung dass durch die (warscheinlich
sch\"arfere)
Eigenschaft $E^{**}$ die halbkompakten Mengen $F_{\sigma}$
allgemein charakterisiert sind.''}
}
\medskip
The existence of a Sierpi\'nski set violates this conjecture. As
we have seen earlier, Sierpi\'nski sets are elements of
$\sone(\ga,\ga)$.
The following result shows that Hurewicz's conjecture fails
in ZFC.
\begin{theorem}\label{nothc}
There exists a separable metric space $X$ such that
$|X|=\omega_1$, $X$ is not $\sigma$-compact and $X$ has property
$\ufin(\ga,\ga)$.
This $X$ also has property $\sone(\ga,\om)$ and $\sfin(\om,\om)$.
\end{theorem}
\pf
\medskip\noindent Case 1. ${\goth b}>\omega_1$.
In this case every $X$ of size $\omega_1$ is in $\sone(\ga,\ga)$ and
$\sfin(\om,\om)$,
hence in both $\ufin(\ga,\ga)$ and $\sone(\ga,\lm)$.
(by Theorems \ref{sfinomega} and \ref{gam3} ).
\medskip\noindent Case 2. ${\goth b}=\omega_1$.
In this case we will use a construction similar to one in
\cite{G-M}.
Build an $\omega_1$-sequence $( x_\alpha:\alpha<\omega_1 )$ of
elements of $[\omega]^\omega$ such that $\alpha < \beta$
implies $x_\beta\subseteq^* x_\alpha$ and if $f_\alpha:\omega\to
x_\alpha$
is the increasing enumeration of $x_\alpha$, then
for every $g\in\oo$ there exists $\alpha$ such that
for infinitely many $n$ we have $g(n)